  For any smooth Riemannian metric on an $(n+1)$-dimensional compact manifold
with boundary $(M,\partial M)$ where $3\leq (n+1)\leq 7$, we establish general
upper bounds for the Morse index of free boundary minimal hypersurfaces
produced by min-max theory in the Almgren-Pitts setting. We apply our Morse
index estimates to prove that for almost every (in the $C^\infty$ Baire sense)
Riemannan metric, the union of all compact, properly embedded free boundary
minimal hypersurfaces is dense in $M$. If $\partial M$ is further assumed to
have a strictly mean convex point, we show the existence of infinitely many
compact, properly embedded free boundary minimal hypersurfaces whose boundaries
are non-empty. Our results prove a conjecture of Yau for generic metrics in the
free boundary setting.
