Spinor-genus correspondence conjecture for ternary quadratic-form genera

Let $G_1$ and $G_2$ be genera of positive ternary forms whose discriminant ratio is an integral squarefree integer, and suppose they admit a genus-correspondence. A genus-correspondence matches forms in the two genera according to mutual representation of the discriminant-ratio multiple and respects spinor genus when two forms in $G_1$ lie in the same spinor genus if and only if their corresponding forms in $G_2$ do.

Spinor-genus correspondence conjecture. Suppose that $G_1$ and $G_2$ each have exactly two spinor genera. Then $G_1$ has spinor exceptional integers if and only if $G_2$ has spinor exceptional integers, and $G_1$ has splitting integers if and only if $G_2$ has splitting integers. Moreover, the genus-correspondence respects spinor genus. When spinor exceptions exist, the regular spinor genera correspond; when splitting integers exist, the spinor genera with larger weighted representation measures for the smallest splitting integers correspond.

The conjecture is supported by extensive numerical evidence and describes how genus-correspondences preserve the spinor-genus structure. It also accounts for the correspondence of regular spinor genera and the relative representation measures associated with splitting integers.