Morales' Ehrhart positivity conjecture for Tesler polytopes

For any $n\geq 1$, let $\operatorname{Tes}_{n}(\boldsymbol{1})$ and $\operatorname{Tes}_{n}(1,0,\dots,0)$ be Tesler polytopes, where $\boldsymbol{1}=(1,\dots,1)$. A polytope is Ehrhart positive if all coefficients of its Ehrhart polynomial are positive. Morales' conjecture. The Tesler polytopes

$$\operatorname{Tes}_{n}(\boldsymbol{1})$$

and

$$\operatorname{Tes}_{n}(1,0,\dots,0)$$

are both Ehrhart positive for every positive integer $n$. This conjecture concerns positivity properties of the lattice-point enumerators of Tesler polytopes and was the initial motivation for the paper; its resolution is not specified in the supplied source.