Sato's conjecture for $\kappa(qp)$

For distinct prime numbers $p$ and $q$, let $H_{p,q}$ be the subgroup of $(\mathbb{Z}/p\mathbb{Z})^{\times}$ spanned by the images of $q$ and $-1$ in $\mathbb{F}_{p}^{\times}$, and let $n_q(p)$ be the index of $H_{p,q}$ in $(\mathbb{Z}/p\mathbb{Z})^{\times}$. Let $\kappa(N)$ denote the dimension of the cokernel of the weight-two map $D_{2}^{\mathrm{iter}}$. Sato's conjecture. For every prime $p\geq5$ and $q\in\{2,3\}$,

$$\kappa(qp)=n_q(p)-1.$$

This conjecture gives an explicit formula for the weight-two obstruction at levels $2p$ and $3p$ in terms of the subgroup generated by $q$ and $-1$ modulo $p$. The source presents it as a numerical conjecture and gives no resolution.