  For given integers $n$ and $d$, both at least 2, we consider a homogeneous
multivariate polynomial $f_d$ of degree $d$ in variables indexed by the edges
of the complete graph on $n$ vertices and coefficients depending on
cardinalities of certain unions of edges. Cardinaels, Borst and Van Leeuwaarden
(arXiv:2005.14566, 2020) asked whether $f_d$, which arises in a model of
job-occupancy in redundancy scheduling, attains its minimum over the standard
simplex at the uniform probability vector. Brosch, Laurent and Steenkamp
(arXiv:2009.04510, 2020) proved that $f_d$ is convex over the standard simplex
if $d=2$ and $d=3$, implying the desired result for these $d$. We give a
symmetry reduction to show that for fixed $d$, the polynomial is convex over
the standard simplex (for all $n\geq 2$) if a constant number of constant
matrices (with size and coefficients independent of $n$) are positive
semidefinite. This result is then used in combination with a computer-assisted
verification to show that the polynomial $f_d$ is convex for $d\leq 9$.
