Windows 10 Home 10.0.19044 Build 19044.
Thunderbird version: 91.11.0 (32-bit)
JAWS Version 2022.2206.9 ILM

Preconditions:
Configure Thunderbird to use Acrobat Reader DC rather than its internal PDF viewer:
In Tools > Preferences > Files and Attachments, for Portable Document Format (PDF) choose Use Adobe Acrobat DC.
inbox contains a message with a file attachment of an untagged PDF document.
JAWS running.
Steps to reproduce:
Open the message.
Open the attached file (Message > Attachments > Open).
Actual: Acrobat Reader appears, JAWS is focused on the document, the JAWS virtual cursor is active and there is no text to read with the virtual cursor. The Reading Untagged Document dialog is not spoken by JAWS.
Expected: the Reading Untagged Document dialog is spoken with focus on the Reading Order combo box.

Note: saving the file and opening it from File Explorer seems to work properly.

Acrobat Reader opens and JAWS is focused on it with the VC active. Inspection with Script Utility Mode (JAWSKey+Windows+NumpadMinus)shows that the dialog appears at the top level (although it is not).
From the initial focus ascend the tree with Shift+F2 to the top level.
Set to speak window class (F3) and tab left encountering:
AVLAVWindow
AVLAVWindow
AcrobatModelessParentWnd Exiting the window tree for the currently active application

32770

Below are node captures (Script Utility Mode ALT+CTL+ENTER) of these last two windows. Note that AcrobatModelessParentWnd has a child window of 0 and prior of AcrobatModelessParentWnd, while #32770 has both parent and next of AcrobatModelessParentWnd.

Node capture for AcrobatModelessParentWnd: {
Handle = 397346 = {61022}
Class = AcrobatModelessParentWnd
Type = Unknown window type
TypeCode = 0
SubtypeCode = 0
ControlID = 0
StyleBits = 0x4c00000
Extended style bits = 0x180
Parent = 0 = {0}
FirstChild = 0 = {0}
Prior = 331862 = {51056}
Next = 1903976 = {1d0d68}
WindowHierarchyX = 128
WindowHierarchyY = 1
WindowRect = (1,1)-(101,101)
controlAttributes = 0
HasTitle = 12582912
WindowDisabled = 0
WindowObscured = 1

WindowVisible = 0

Highlighted text:

All Text Restricted:

All Text Inclusive:

} ; AcrobatModelessParentWnd

32770: {

Handle = 331862 = {51056}
Class = #32770
Type = dialog
TypeCode = 18
SubtypeCode = 18
ControlID = 0
StyleBits = 0x94c80000
Extended style bits = 0x180
Parent = 397346 = {61022}
FirstChild = 790170 = {c0e9a}
Prior = 4132804 = {3f0fc4}
Next = 397346 = {61022}
WindowHierarchyX = 127
WindowHierarchyY = 2
WindowRect = (429,126)-(939,605)
controlAttributes = 0
HasTitle = 12582912
WindowDisabled = 0
WindowObscured = 1

WindowVisible = 1

Highlighted text:

All Text Restricted:

Reading Untagged Document with Assistive Technology

All Text Inclusive:
Reading Untagged Document with Assistive Technology An assistive technology like a screen reader may be running on your machine. This untagged, 2-page
document must be prepared for reading with assistive technology. If you are not using an assistive technology, check the "Do not show this dialog again"
box below. Reading Order Options Reading Order: Infer reading order from document (recommended) Reading Mode Options Read the currently visible pages only
Read the entire document Do not show this dialog again. Use the settings from the Reading Preferences. Start Cancel

} ; #32770

Acrobat Reader Version:
Continuous Release | Version 2022.001.20169 | 64-bit
Version File Version: 22.001.20169.0
| 64-bit
AGM Version 4.030.00119 | 64-bit
CoolType Version 5.017.00013 | 64-bit
Core Version 22.256 | 64-bit
JP2K Version 2.000.00006.50420 | 64-bit

Work-around:
When Reader appears, try to Alt+tab away and back. If this doesn't work,
Activate Touch cursor with Advanced mode on
Arrow up until it says "window"
Arrow left, it says "...reading untagged..."
Arrow down twice, it says "an assistive..."
Arrow right until you reach Start button
Press Space.

Did not try this with advanced mode off, might be easier.

From Generate System Report (opened Reader from Desktop, not from Thunderbird): {
Account Detail:
User Rights: Admin
User Account Control: Limited
Process Integrity: Medium
Profile Type: None
Acrobat Detail:
Sandboxing: Off
Protected View: Off
Captive Reader: No
Multi-Reader on Desktop Support: Off
Applications Installed:
Available Physical Memory: 1415584 KB
Available Virtual Memory: 135286863632 KB
BIOS Version: LENOVO - 1
Default Browser:
Default Mail:
Display Detail:
Screen Width: 1366
Screen Height: 768
Number of Monitors: 1
Number of Mouse Buttons: 2
Has Mouse Wheel: No
Has Pen Windows: No
Double Byte Character Set: No
Has Input Method Editor: Yes
Inside Screen Reader: Yes
Graphics Card: Intel(R) HD Graphics 4600
Version: 20.19.15.4624
Check: Not Supported
Installed Acrobat: C:\Program Files\Adobe\Acrobat DC\Acrobat\Acrobat.exe
Version: 22.1.20169.0
Creation Date: 2022/07/04
Creation Time: 10:34:20 PM
Locale: English (United States)
Monitor:
Name: Freedom Scientific Accessibility Display Driver
Resolution: 1366 x 768 x 0
Bits per pixel: 32
Monitor:
Name: Intel(R) HD Graphics 4600
Resolution: 1366 x 768 x 60
Bits per pixel: 32
OS Manufacturer: Microsoft Corporation
OS Name: Microsoft Windows 10
OS Version: 10.0.19044

Page File Space: 11444504 KB
Processor: Intel64 Family 6 Model 60 Stepping 3 GenuineIntel ~2494 Mhz
Session Detail:
Boot Type: Normal
Is Shutting Down: No
Network: Available
Inside Citrix: No
Inside VMWare: No
Remote Session: No
Remote Control: No
Using JAWS: Yes
Using ZoomText: No
Using Windows-Eyes: No
Using NVDA: No
System Name: GARYSPC
Temporary Directory: C:\Users\Gary\AppData\Local\Temp\
Time Zone: Mountain Standard Time
Total Physical Memory: 6201624 KB
Total Virtual Memory: 137438953344 KB
User Name: Gary
Windows Detail:
Tablet PC: Yes
Starter Edition: No
Media Center Edition: No
Slow Machine: No
Windows Directory: C:\WINDOWS
} ; 1
2 {
C:\Program Files\Adobe\Acrobat DC\Acrobat\plugins\Accessibility.api
Version: 22.1.20169.0
Creation Date: 2022/07/04
Creation Time: 10:34:22 PM
C:\Program Files\Adobe\Acrobat DC\Acrobat\plugins\Annots.api
Version: 22.1.20169.0
Creation Date: 2022/07/04
Creation Time: 10:34:22 PM
C:\Program Files\Adobe\Acrobat DC\Acrobat\plugins\IA32.api
Version: 22.1.20169.0
Creation Date: 2022/07/04
Creation Time: 10:34:22 PM
C:\Program Files\Adobe\Acrobat DC\Acrobat\plugins\Updater.api
Version: 22.1.20169.0
Creation Date: 2022/07/04
Creation Time: 10:34:22 PM
} ; 2