Automorphism-action compatibility conjecture for analytic K3 surfaces

Let $X^{an}/K$ be an analytic K3 surface with the lattice $\Lambda_X$, period homomorphism $\rho_X$, valuation vector $v_X:=val_K\circ\rho_X$, and affine piecewise-linear action on $S^2$. Let $\Gamma_{\rho_X}$ be the image of $Aut(X)$ in $Aut(\Lambda_X,\rho_X)$ and let $\phi_{v_X}$ be the homomorphism to $Aut_{{\bf Z}PL,v_X}(S^2)$ defined by the preceding monodromy conjecture. Automorphism-action compatibility conjecture. The group $\Gamma_{\rho_X}$ is a subgroup of $\Gamma_{v_X}$, and the homomorphism $Aut(X)\to Aut_{{\bf Z}PL,v_X}(S^2)$ is conjugate to the restriction of $\phi_{v_X}$ to $\Gamma_{\rho_X}$. This predicts compatibility between K3 automorphisms, period data, and the induced affine action; it remains open in the source.