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 Finally, using an unpublished result of Gabber, we also show that $F$-injective and Du Bois singularities have a common definition in terms of smooth hypercovers.

v3, the proof of Lemma 3.9, step (2) was incorrect, this version corrects the error. The statement of Lemma 3.9 is unchanged. Remark 3.10 is added which explains the issue