Monotonicity conjecture for the binomial-logarithm sequence

Let $c_p$ and $w_p$ be the quantities defined by

$$c_{p}=-\frac{1}{p}\sum_{k=1}^{p}(-1)^{k}\binom{p}{k}\ln k,\qquad w_{p}=-\frac{1}{p}\sum_{k=1}^{p}\binom{p}{k}(-1)^{k}\left(\ln k\right)^{2}.$$

Define the sequence

$$a_p:=\frac{\pi^{2}}{6}+p w_p-p^{2}c_p^{2}.$$

Monotonicity conjecture. The sequence $a_p$ is decreasing in $p$. The preceding asymptotic result gives $a_p\to 0$ as $p\to\infty$, so this conjecture would describe the finite-$p$ approach to the limiting constant $\pi^2/6$.