Quasi-polarized isogeny realization conjecture for K3 surfaces

Let $(X,\xi)$ be a quasi-polarized K3 surface over $k$. Let $(\Lambda_p,\lambda_p)$ and $(\Lambda^p,\lambda^p)$ be pointed lattices over $W(k)$ and $\hat{\mathbb{Z}}^p$, respectively, with underlying lattices abstractly isomorphic to $H^2_{\mathrm{cris}}(X/W(k))$ and $H^2_{\mathrm{\acute et}}(X,\hat{\mathbb{Z}}^p)$. Equip $\Lambda_p$ with a Frobenius action $\varphi$ such that $(\Lambda_p,\varphi)$ is a K3 crystal. Quasi-polarized isogeny realization conjecture. For each pair of isometric embeddings

$$((\Lambda_p,\lambda_p),\varphi)\subset ((H^2_{\mathrm{cris}}(X/W(k))[1/p],c_1(\xi)),F),$$

and

$$(\Lambda^p,\lambda^p)\subset (H^2_{\mathrm{\acute et}}(X,\mathbb{A}_f^p),c_1(\xi)),$$

there exists a quasi-polarized K3 surface $(X',\xi')$ and an isogeny $f:X\rightsquigarrow X'$ such that

$$f^*(H^2_{\mathrm{cris}}(X'/W(k)),c_1(\xi'))=(\Lambda_p,\lambda_p),$$

and

$$f^*(H^2_{\mathrm{\acute et}}(X',\hat{\mathbb{Z}}^p),c_1(\xi'))=(\Lambda^p,\lambda^p).$$

This is the quasi-polarized refinement of the preceding realization conjecture, formulated using pointed lattices; the source gives no resolution.