Support provided this fix. Maybe it won't help everybody here but it's worth a try.
Even I was able to apply it.
There is a bug in CS-Cart < 4.9.2 version related to page headers, when script couldn't parse pages with HTTP/2 headers. In order to fix it, please open file app/Tygh/Http.php and replace this line:
while (strpos(ltrim($content), 'HTTP/1') === 0) {
with this one:
while (strpos(ltrim($content), 'HTTP/') === 0) {
Then save the file and check the upgrade function.