Generalized Turán inequality conjecture for plane-partition polynomials

Let $P_n(x)$ be the polynomials associated with generalized plane partitions, and define

$$\Delta_{a,b}(x):=P_{a-1}(x)P_{b+1}(x)-P_a(x)P_b(x).$$

Let $a$ and $b$ be integers satisfying $a-1>b\geq 0$ and $(a,b)\notin\{(4,2),(6,4)\}$. Generalized Turán inequality conjecture. For every real number $x\geq 2$,

$$\Delta_{a,b}(x)>0.$$

This generalizes the preceding Turán inequality and the Chern–Fu–Tang conjecture for $k$-colored partitions. The source provides the formulation and motivation but does not report a proof in the stated generality.