I gave up after the response I got from A1SG, I'll manually change the date. My provider will have no clue what I'm talking about.
I know what you mean - the support desk of some providers is "optimizable"

But they should know at least, what "last modified date" means.
Which provider is this? <- For avoiding in future :-)
If you can change the source code of your files you could apply this php-code snippet
header("Expires: " . $lastmod_gm); // sende echtes Datum
header("Last-Modified: " . $lastmod_gm);
$moddate = gmdate("d.m.Y", $lastmod_mk); /* f. copyright im Seiten-Footer */
I use it dynamically (supplemented by additional functions) to tell Google newer dates then the saving dates of the files probably do

But I didn't implement some code for the directories so far.