Working on my SEO, I discovered that the site description on the header is H2 and not H1.
As a consequence, there is no H1 on the main page, at the exception of the sliders (which is quite another problem since having multiple H1 is very bad for SEO).
What do you think of my proposal for a further patch :
-Switching from H1 to H2 for sliders title
-Switching from H2 to H1 for site description on main page header
-Keeping H2 for site description on other pages