@mahesh
Many thanks, Mahesh!
I still have a lot to learn (in the sticky posts, for instance, I had no idea what I was doing…), but your instructions made everything work! 
By the way, in the end I also added an extra chunk of custom CSS to remove the space on top of the header (maybe someone else finds it useful):
@media screen and (min-width: 1061px) {
	.site { position:center; }
	#masthead {
		background-color: none;
		display: block;
		left: 0;
		position: fixed;
		top: 0;
		width:100%;
		z-index: 999;
	}
	#hgroup-wrap,
	#header-menu {
		margin: 0 auto;
		width: 0px;
	}
Thanks again! 😉