Turán inequality conjecture for the plane-partition polynomials

Let $P_n(x)$ denote the polynomials associated with the generalized plane partitions, and let $a$ be an integer with $a\geq 12$. Turán inequality conjecture. For every real number $x\geq 1$,

$$P_a(x)^2>P_{a-1}(x)\,P_{a+1}(x).$$

This is the polynomial analogue of log-concavity for plane partition numbers and is named a Turán inequality in the source. Numerical zero-distribution evidence verifies the inequality for $12\leq a\leq 100$, while the general assertion remains open.