Twisted derived Torelli conjecture for K3 surfaces in positive characteristic

Let $S$) and $S'$ be K3 surfaces over a finite field $k$ with $\operatorname{char}(k)=p>0$. A prime-to-$p$ derived isogeny is an equivalence of derived categories arising from a prime-to-$p$ cohomological isogeny, and a prime-to-$p$ isogeny is an isogeny whose crystalline realization is integral. Twisted derived Torelli conjecture. The following are equivalent:

1. There exists a prime-to-$p$ derived isogeny $D^b(S)\sim D^b(S')$.
2. There exists a prime-to-$p$ isogeny between $S$ and $S'$.

This is presented as a formulation of the twisted derived Torelli conjecture for K3 surfaces. The implication from a prime-to-$p$ derived isogeny to a prime-to-$p$ isogeny is clear, while the converse remains open.