Vendi's conjecture that Vendi Score bounds LCR

Let $Z$ be an $n\times n$ positive semidefinite similarity matrix with $n>0$, symmetric entries in $[0,1]$, and diagonal entries equal to $1$. On the uniform distribution, write LCR at order $q$ as $D_q(Z, p=\frac{1}{n})$ and Vendi Score as $\operatorname{VS}_q(Z)$. Vendi's conjecture. For every $q\in[-\infty,+\infty]$,

$$\operatorname{VS}_q(Z)\geq D_q\left(Z,\frac{1}{n}\right).$$

The conjecture formalizes the empirical observation that Vendi Score is consistently at least as large as LCR, with both measures approaching $1$ as the half-distance decreases. The cases $q=3$ and $q=\infty$ are proved in the paper; the general case remains open, including the unresolved case $q=1$.