  For $\ba \in \R_{\geq 0}^{n}$, the Tesler polytope $\tes_{n}(\ba)$ is the set
of upper triangular matrices with non-negative entries whose hook sum vector is
$\ba$. Motivated by a conjecture of Morales', we study the questions of whether
the coefficients of the Ehrhart polynomial of $\tes_n(1,1,\dots,1)$ are
positive. We attack this problem by studying a certain function constructed by
Berline-Vergne and its values on faces of a unimodularly equivalent copy of
$\tes_n(1,1,\dots,1).$ We develop a method of obtaining the dot products
appeared in formulas for computing Berline-Vergne's function directly from
facet normal vectors. Using this method together with known formulas, we are
able to show Berline-Vergne's function has positive values on codimension $2$
and $3$ faces of the polytopes we consider. As a consequence, we prove that the
$3$rd and $4$th coefficients of the Ehrhart polynomial of $\tes_{n}(1,\dots,1)$
are positive. Using the Reduction Theorem by Castillo and the second author, we
generalize the above result to all deformations of $\tes_{n}(1,\dots,1)$
including all the integral Tesler polytopes.
