  Heim, Neuhauser, and Tr\"oger recently established some inequalities for
MacMahon's plane partition function $\mathrm{PL}(n)$ that generalize known
results for Euler's partition function $p(n)$. They also conjectured that
$\mathrm{PL}(n)$ is log-concave for all $n\geq 12.$ We prove this conjecture.
Moreover, for every $d\geq 1$, we prove their speculation that $\mathrm{PL}(n)$
satisfies the degree $d$ Tur\'an inequality for sufficiently large $n$. The
case where $d=2$ is the case of log-concavity.
