Spinor-genus preservation conjecture for genus-correspondences

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 respects spinor genus when corresponding forms lie in the same spinor genus on one side if and only if they lie in the same spinor genus on the other side.

Spinor-genus preservation conjecture. Suppose that $G_1$ and $G_2$ have exactly the same number, namely some power $2^j$, of spinor genera. Then the genus-correspondence respects spinor genus.

This extends the preceding two-spinor-genus prediction to genera with four or more spinor genera, but the source explicitly says that it is offered with less detail and far less evidence.