reading an environment variable from screen

When I reattach to screen, the SSH_AUTH_SOCK is wrong.

I often ssh in and run screen simultaneously on a laptop and desktop.  So I can’t use the trick of symlinking something to /tmp/ssh-xxxxxxxxxx.

Is there some way to interrogate screen to tell me what the contents of an environment variable are?

Running this in a shell inside of screen:

       screen -S $STY -X colon 'exec |echo $SSH_AUTH_SOCK\r'

gives me the info I’m looking for, but it pushes it out to the display.  I’m so close!  How can I capture this output?

My goal here is to create a sort of ssh/scp pre-command as a shell alias that probes screen each time and sets the SSH_AUTH_SOCK variable correctly.