Factorization conjecture for shapes of tridiagonal pairs

Let $(A,A^*)$ be a tridiagonal pair over $\mathbb{K}$ with diameter $d$, and let $\{\rho_i\}_{i=0}^d$ denote its shape, where $\rho_i$ is the corresponding shape multiplicity. Shape-factorization conjecture. There exists a nonnegative integer $N$ and positive integers $d_1,d_2,\ldots,d_N$ such that

$$\sum_{i=0}^d \rho_i z^i=\rho_0\prod_{j=1}^N(1+z+z^2+\cdots+z^{d_j}).$$

Here $z$ is an indeterminate. The factorization would impose a strong combinatorial structure on the shape of every tridiagonal pair; the source gives no resolution of this conjecture.