The vanishing conjecture for $\kappa(2^a3^b)$

For $N\in\mathbb{Z}_{\geq1}$, let $\kappa(N)$ be the dimension of the cokernel of the weight-two map $D_{2}^{\mathrm{iter}}$. Let $a,b\in\mathbb{Z}_{\geq0}$. Vanishing conjecture for $\kappa(2^a3^b)$.

$$\kappa(2^{a}3^{b})=0.$$

This predicts vanishing of the obstruction measured by $\kappa$ for every level whose prime divisors are among $2$ and $3$. The statement is based on numerical results; the source proves it when $a=0$ or $b=0$, but gives no resolution in the remaining cases.