The cover enumerator conjecture for type-B parabolic Tamari lattices

Let $t>0$ and let $\alpha=(t,1,1,\ldots,1)$ be a type-$B$ composition of $n$. Define the cover enumerator

$$c_{\alpha}(x)=\sum_{\pi\in\mathfrak{H}_{\alpha}(231)}x^{\lvert\operatorname{Cov}(\pi)\rvert}.$$

Cover enumerator conjecture.

$$c_{\alpha}(x)=\sum_{k=0}^{n-t}\binom{n-t}{k}\binom{n+t}{k}x^k.$$

Consequently,

$$\left\lvert\mathfrak{H}_{\alpha}(231)\right\rvert=\binom{2n}{n-t}.$$

This is proposed from computer experiments as an enumerative formula for the type-$B$ parabolic Tamari lattice associated with $\alpha$; its status is not resolved in the supplied text.