  Recently M. Mustata and V. Srinivas related a natural conjecture about the
Frobenius action on the cohomology of the structure sheaf after reduction to
characteristic $p > 0$ with another conjecture connecting multiplier ideals and
test ideals. We generalize this relation to the case of singular ambient
varieties. Additionally, we connect these results to a conjecture relating
$F$-injective and Du Bois singularities. Finally, using an unpublished result
of Gabber, we also show that $F$-injective and Du Bois singularities have a
common definition in terms of smooth hypercovers.
