Schwede's Du Bois and dense F-injective type conjecture

Let $X$ be a reduced scheme of finite type over an algebraically closed field of characteristic $0$. A scheme is of dense $F$-injective type if it has reductions to characteristic $p>0$ that are $F$-injective for a Zariski-dense set of primes. Schwede's conjecture. $X$ has Du Bois singularities if and only if $X$ is of dense $F$-injective type. This conjecture proposes an equivalence between a characteristic-$0$ condition and the corresponding dense-characteristic-$p$ condition; the source records it as a conjecture, without resolving it.