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This is not possible. Scripts get activated by Greasemonkey/Tampermonkey after the page is loaded, by design.
It does not require to have the page fully loaded to do things on it. As soon as the overlay is loaded the script can take care of it. Many other scripts do this. GoogleMonkeyR, Hit Hider by domain, Text to link just to name a few.
Delay
I noticed that the script is working nicely but the overlay disappears after the page fully finishes loading could that happen earlier?