Just found what appears to be a solution for this. I have eFax Messenger Plus 3.0.1.9 installed, so your mileage may vary.

I went to the eFax Messenger Plus Start Menu group. Under Messenger Tools, I choose Live Menu Settings. Under Live Menu options, I moved Acrobat Reader 4.0,5.0 and Adobe Acrobat 4.0, 5.0 to the Hide menu side.

Just to confirm, I am using Adobe Acrobat 6.0...not just reader, and not the 5.0 versions.

This APPEARS to have worked for me. I was able to quickly test by opening multiple documents and my file menu never went away. I "unhid" these apps, and my file menu disappeared again.

Let's have some others try it out and see if this solves it!