Conjecture on the factorization of constant-term polynomials for the Hecke group

Let $\overline{h}_k(x)=\nu_k\,p_{k,1}(x)\,p_{k,2}(x)\cdots p_{k,\alpha}(x)=\nu_k\,\widetilde{p}_k(x)$, where each $p_{k,n}$ is monic and $\nu_k$ is rational. Write the OEIS sequence $\{0,1,47,2488,138799,\ldots\}$ as $\{a_0,a_1,\ldots\}$, and let $d_2(k)$ and $d_3(k)$ denote the relevant exponent functions used in the paper. Factorization conjecture. The following identities and divisibility properties hold:

$$\nu_k=24a_k.$$ $$\widetilde{p}_k(3)\text{ is always odd}.$$ $$\operatorname{ord}_2(a_k)=3d_2(k)-3.$$ $$\operatorname{ord}_3(\widetilde{p}_k(3))=d_3(k)-1.$$

These assertions summarize observed divisibility patterns in the constant terms $A_{\overline{\mathcal{K}},k,m}(0)$ and their associated polynomials. The supplied text does not establish them or provide evidence of resolution, so their status remains open.