Higher-order Turán inequality for the partition function

Let $p(n)$ denote the number of partitions of the positive integer $n$, and define

$$u_n=\frac{p(n+1)p(n-1)}{p(n)^2}.$$

Higher-order partition-function inequality. For $n\geq 2$,

$$4(1-\nu_n)(1-\nu_{n+1})<\left(1+\frac{\pi}{\sqrt{24}n^{3/2}}\right)(1-\nu_n\nu_{n+1})^2.$$

This is presented as a higher-order analogue of a conjecture of DeSalvo and Pak. The supplied text does not state that this inequality has been proved or disproved, so its resolution is unclear.