Monodromy factorization conjecture for K3 affine structures

Let ${\cal F}_v$ be a leaf indexed by a positive vector $v$ in the K3 period description, let $\Gamma_v$ be the corresponding stabilizer group, and let $Aut_{{\bf Z}PL,v}(S^2)$ be the group of piecewise-linear transformations of $S^2$ with integer linear part. Monodromy factorization conjecture. The monodromy homomorphism $\pi_1({\cal F}_v)\to Aut_{{\bf Z}PL,v}(S^2)$ factors as

$$\pi_1({\cal F}_v)\twoheadrightarrow\Gamma_v\xrightarrow{\phi_v}Aut_{{\bf Z}PL,v}(S^2),$$

where $\phi_v$ is uniquely determined by this property. This conjecture identifies the affine monodromy of the leaf with the arithmetic stabilizer action; it remains open in the source.