Kernel-dimension conjecture for degree-two theta series of rank-six lattices

Let $G_p$ be the genus of lattices of rank $6$ and discriminant $p$, and let $k_p$ be the number of isomorphism classes in $G_p$ of lattices with no automorphism of determinant $-1$. Kernel-dimension conjecture. The kernel of $\theta^{(2)}$ on $G_p$ has dimension $k_p$. This conjecture concerns the unexpectedly frequent failure of injectivity of the degree-two theta map in rank $6$; the surrounding computations provide evidence, but no proof or resolution is given.