Polynomial interpolation conjecture for constant terms of Hecke-group modular forms

Let $\mathcal{K}$ and $\overline{\mathcal{K}}$ be the two families of meromorphic modular forms in the source, and let $A_{\mathcal{K},k,m}(0)$ and $A_{\overline{\mathcal{K}},k,m}(0)$ denote their constant terms. Let $Q_{\mathcal{K},k,0}(x)$ and $Q_{\overline{\mathcal{K}},k,0}(x)$ be the corresponding polynomial functions, and let $h_k(x)$ and $\overline{h}_k(x)$ be the polynomials obtained by interpolation from these constant terms. Polynomial interpolation conjecture. The identities

$$h_k(x)\equiv Q_{\mathcal{K},k,0}(x),\qquad h_k(m)=A_{\mathcal{K},k,m}(0)$$

and

$$\overline{h}_k(x)\equiv Q_{\overline{\mathcal{K}},k,0}(x),\qquad \overline{h}_k(m)=A_{\overline{\mathcal{K}},k,m}(0)$$

hold identically. The authors describe this as their fundamental assumption that the observed linearity behavior is reliable; the conjecture is used as the basis for subsequent empirical divisibility predictions.