Terwilliger systems' rank-generating polynomial factorization conjecture

Let $\Phi$ be a TD system with diameter $d$, and let $\rho_0,\rho_1,\ldots,\rho_d$ be the corresponding scalars from Corollary 3.4. Consider the polynomial

$$P(t)=\sum_{i=0}^d\rho_i t^i$$

in the variable $t$.

Terwilliger systems' rank-generating polynomial factorization conjecture. There exist positive integers $d_1,d_2,\ldots,d_n$ whose sum is $d$ such that

$$P(t)=(1+t+t^2+\cdots+t^{d_1})(1+t+t^2+\cdots+t^{d_2})\cdots(1+t+t^2+\cdots+t^{d_n}).$$

This conjecture predicts a product decomposition for the polynomial formed from the associated scalars of a TD system. The supplied source context does not state a proof or disproof, so its general status remains open.