Uniqueness conjecture for lattices associated with symplectic K3 degenerations

Let $H$ be a finite symplectic automorphism group of a Kählerian K3 surface, with abstract group type indexed by $n$, and let the degeneration type be specified by a Dynkin diagram. For a Kählerian K3 surface $X$ with $S_X<0$, let $P(X)$ be the set of classes of nonsingular rational curves on $X$, let

$$N_H=((S_X)^H)^{\perp}_{S_X}$$

be the coinvariant sublattice, and suppose that $S_X$ is generated by $N_H$ and $P(X)$ up to finite index, with

$$\operatorname{rk} S_X=\operatorname{rk} N_H+1.$$

Uniqueness conjecture. For a fixed abstract group type of $H$ (equivalently, a fixed $n$) and a fixed Dynkin-diagram type of degeneration, the corresponding lattice

$$S=[N_H,\alpha_{i,j}]_{pr}$$

is unique up to isomorphism. Equivalently, the isomorphism class of $S=S_X$ is uniquely determined by the abstract group type of $H$ and the Dynkin-diagram type of $P(X)$, except in exactly two cases: $n=34$, equivalently $H\cong \mathfrak S_4$, with degeneration type $6\mathbb A_1$, and $n=10$, equivalently $H\cong D_8$, with degeneration type $2\mathbb A_1$. In each exceptional case there are exactly two isomorphism classes of lattices $S$. This conjecture refines the preceding result that the discriminant group of the corresponding lattice is determined by the group and degeneration types; the two stated exceptions are the only failures of uniqueness of the lattice itself.