Kernel-dimension conjecture for degree-two theta series of prime-discriminant rank-six genera

Let $G_p$ be the genus of lattices of rank $6$ and discriminant $D=p$. Kernel-dimension conjecture. The kernel of $\theta^{(2)}$ on $G_p$ has dimension equal to the number of classes in $G_p$ of lattices with no automorphism of determinant $-1$. This is a computationally motivated formulation of the same rank-six prime-discriminant phenomenon as the fuller version stated elsewhere in the paper; the source gives no evidence that it has been proved.