Polynomiality of coefficients in the covariant derivatives of eigenfunctions

Let $a^{i_1i_3\cdots i_{2l-1}}_{i_2i_4\cdots i_{2l}}$ be the coefficients arising in the covariant derivatives of an eigenfunction along parallel tensors, and let $\theta$ denote the eigenvalue parameter. The ambient space has dimension $n$, sectional curvature is $K$, and the target polynomial has the form

$$C \prod_{p=0}^{k-1}(\theta+Kp(n+p-1))=C\theta(\theta+Kn)\cdots(\theta+K(k-1)(n+k-2)),$$

where $C$ depends only on $n$.

Polynomiality conjecture. Each coefficient $a^{i_1i_3\cdots i_{2l-1}}_{i_2i_4\cdots i_{2l}}$ is a polynomial in $\theta$ of degree $l$. Conjecturally, the constant in the displayed factorization is

$$C=\frac{(n-1)n(n+1)\cdots(n+r-2)}{n(n+2)(n+4)\cdots(n+2r-2)}.$$

This conjecture concerns the polynomial structure underlying covariant derivatives of eigenfunctions on space forms and is motivated by the vertex-algebraic structure discussed in the paper. The supplied excerpt does not establish the polynomiality or the proposed formula for $C$, so their resolution remains unclear.