Genus-correspondences respecting spinor genus

For two positive definite integral ternary quadratic forms f and g and a positive integer n, if n⋅g is represented by f and n⋅dg=df, then the pair (f,g) is called a representable pair by scaling n. The set of all representable pairs in gen(f)×gen(g) is called a genus-correspondence. In [6], Jagy conjectured that if n is square free and the number of spinor genera in the genus of f equals to the number of spinor genera in the genus of g, then such a genus-correspondence respects spinor genus in the sense that for any representable pairs (f,g),(f′,g′) by scaling n, f′∈spn(f) if and only if g′∈spn(g). In this article, we show that by giving a counter example, Jagy's conjecture does not hold. Furthermore, we provide a necessary and sufficient condition for a genus-correspondence to respect spinor genus.