Equivalent trace and row-sum bounds for Vendi Score and LCR

Let $Z$ be an $n\times n$ positive semidefinite similarity matrix as above, and let $\mathbf{1}$ denote the $n$-dimensional all-ones vector. Equivalent bounds conjecture. For $q>1$,

$$\operatorname{tr}(Z^q)\leq\sum_i\left(\sum_j Z_{ij}\right)^{q-1}=\mathbf{1}^{\mathsf T}(Z\mathbf{1})^{q-1},$$

while for $q<1$,

$$\operatorname{tr}(Z^q)\geq\mathbf{1}^{\mathsf T}(Z\mathbf{1})^{q-1}.$$

These inequalities would imply the Vendi Score–LCR bound for all orders other than $q=1$, potentially with the missing case obtained by continuity. They are presented as an approach to proving the preceding conjecture and are not established in the source.