By default, Photo Mechanic writes XMP metadata such as star ratings to XMP sidecar files for most RAW files.
When you open an XMP sidecar file in a text editor, you should see the following when the image is assigned a star rating in Photo Mechanic:
xmp:Rating='star_rating_number_here'
Star Ratings will not appear in Lightroom Classic's Import window, however, they should be displayed in Lightroom Classic's Library and Develop modules.
If star ratings were assigned to images in Photo Mechanic after they were imported into Lightroom Classic's catalog, you may need to tell Lightroom Classic to read metadata from file.
Lightroom Classic should show an icon at the top right of a thumbnail indicating that metadata was changed externally and needs to be updated. More information can be found in the "Resolve metadata conflicts between Lightroom Classic, Adobe Bridge, and Camera Raw" section of the following Adobe article:
https://helpx.adobe.com/lightroom-classic/help/advanced-metadata-actions.html#resolve_metadata_conflicts_between_lightroom_adobe_bridge_and_camera_rawIn this case, you would choose "Import Settings From Disk".
Note: this will remove any color edits saved in Lightroom Classic's database that were not previously saved directly to XMP metadata.
While less common, another possibility is that images are being saved to a drive with a case-sensitive filesystem.
Lightroom Classic seems to use lowercase file extensions, so you would need to change the following settings in Photo Mechanic to tell it to use lowercase extensions when creating new files.
In Photo Mechanic navigate to
Preferences-->Files, then in the "File extensions" dropdown, select the "Use lowercase extensions" option.
Note: Changing this setting only affects files created after the setting is changed.