Heim–Neuhauser–Tröger's log-concavity conjecture for the plane partition function

Let $\mathrm{PL}(n)$ denote the number of plane partitions of size $n$. A sequence is log-concave if $\mathrm{PL}(n)^2\geq \mathrm{PL}(n-1)\mathrm{PL}(n+1)$ whenever the relevant indices are defined.

Heim–Neuhauser–Tröger's conjecture. The function $\mathrm{PL}(n)$ is log-concave for every integer $n\geq 12$.

Heim, Neuhauser, and Tröger proved log-concavity for sufficiently large $n$; the conjecture asserts the explicit threshold $n=12$. The present paper proves this conjecture.