The shape-vector binomial bound for tridiagonal pairs

Let $A,A^*$ be a tridiagonal pair on a finite-dimensional vector space $V$ over an algebraically closed field $\mathbb K$. Let $d$ be its diameter, and let $(\rho_0,\rho_1,\ldots,\rho_d)$ be its shape vector, where $\rho_i$ is the common dimension of the corresponding eigenspaces of $A$ and $A^*$. The shape-vector binomial bound. For $0\leq i\leq d$, one has

$$\rho_i\leq \binom{d}{i}.$$

The shape vector is already known to be symmetric and unimodal; this conjecture gives a sharper universal upper bound for its entries. Its status is not resolved in the supplied source.