Valuation conjecture for the K-family at prime parameters

Let $A_{\mathcal K,k,m}(0)$ denote the constant term for the second family of meromorphic modular forms. K-family prime-parameter conjecture. The following assertions hold: if $p>2$ is prime, then

$$\operatorname{ord}_p\left(A_{\mathcal K,p,p}(0)\right)=-2-2p;$$

the value of $\operatorname{ord}_p(A_{\mathcal K,p,p^n}(0))$ is independent of $n=2,3,\ldots$; and if $n>1$ is an integer and $d_2(p)>2$, then

$$\operatorname{ord}_p\left(A_{\mathcal K,p,p^n}(0)\right)=-2-2p.$$

These are data-driven assertions supported by files linked from the authors' repository; the source does not establish them theoretically.