Weight-string formula for the shape of a tridiagonal pair

Let $w_1,w_2,\ldots,w_d$ be the weights associated with the tridiagonal-pair construction, and let $q$ be the scalar used to define a $q$-string, namely a set of the form $w,wq^2,wq^4,\ldots,wq^{2n}$. Assume either $d=3$ or the commutativity conjecture for $LR,L^2R^2,\ldots,L^dR^d$ holds. Decompose the weights as a union of $q$-strings in which every two strings are in general position; let $n$ be the number of strings and $d_1,\ldots,d_n$ their lengths. Weight-string shape conjecture. Then

$$\sum_{i=0}^d\rho_i t^i=\prod_{k=1}^n(1+t+t^2+\cdots+t^{d_k}).$$

For $d=3$, the surrounding text supplies special-case evidence; the general assertion depends on the preceding commutativity conjecture and is not resolved in the supplied text.