...should that change the image I see in the contact sheet...
The simple answer is no, PM will not show the edited image. (I wish you had spelled out the type of RAW-image, all RAW-files are not created equal)
A typical image file contains more than one set of pixel data, and sets may even be different image formats.
RAW is ambiguous these days, some people tend to think a DNG-file is always also a RAW-file.
A typical RAW-file contains multiple sets of "preview" images in addition to the raw image. PM displays preview images embedded in the raw file. PM can not display the raw image data. Well, PM could, but you would not see what you expected. Raw pixel data must be "developed" into a viewable image. PM does not do that, and there is no need as the camera already did and embedded the results in the RAW-file.
To make edits visible to PM, your image edit software would have to replace the embedded preview images. My old trusty Nikon Capture-NX2 actually did that, and Nikon was peppered with comments on slow software. Today's software take pride in
never changing the RAW-file. It is the non-destructive-edit paradigm. If the RAW-file never changes, you will always see the same previews in PM, right?
DNG-files make the exception. Some RAW-file processors allow you to update embedded previews in DNG-files.
If your RAW-files are not DNGs, your best option is to save a jpeg file that reflects your edits and put it along the RAW-file. Make sure the name of each jpeg file matches the name of the corresponding RAW-file, except for the file name extension. PM recognizes image file pairs of one RAW-file and one JPG-file and lets you decide if the contact sheet should display the RAW-files (embedded preview), the JPG-files (with your edits) or both.
I hope this helps.