Hi Kornél
Admittedly we are currently unable to replicate this issue. Switching the menus is controlled by browser side CSS media queries so it would be safe to assume that this is not a server side issue.
A bit of a shot in the dark but could you try adding the following to the Custom CSS field of your templates parameters...
@media only screen and (max-width: 1140px) {
.hornav > ul {display:none !important;}
.slicknav_menu {display:block;}
}
Failing that may I ask does refreshing the page at the split screen width resolve the issue?
Ciarán