Valuation conjecture for norms of singular moduli of discriminant $-3f^2$

Let $p$ be a prime number, let $v_p:4^*\to\mathbb{Z}$ be the usual $p$-adic valuation, and let $D=-3f^2$ be a discriminant. Let $j_D$ be any singular modulus relative to $D$, and let $\mathbb{Q}(j)$ denote the field generated by $j_D$. Valuation conjecture. If $f=p^n$ with $p$ an odd prime, then

$$v_p\left(N_{\mathbb{Q}(j)/\mathbb{Q}}(j_D)\right)=1.$$

If $f=2^n$, then

$$v_2\left(N_{\mathbb{Q}(j)/\mathbb{Q}}(j_D)\right)=4.$$

The conjecture is suggested by the explicit factorizations of norms for conductors that are powers of $2$ and by the preceding arguments for odd prime conductors; its general validity is not established in the supplied text.