Conjecture on the linear coefficient of the Grassmannian Eulerian polynomial

For integers $k$ and $n$ in the range where the polynomial $\hat{E}_{k,n}(q)$ is defined, let $[q]\hat{E}_{k,n}(q)$ denote the coefficient of $q$ in $\hat{E}_{k,n}(q)$. Linear-coefficient conjecture.

$$[q]\hat{E}_{k,n}(q) = {n \choose k+1}{n \choose k-2}.$$

The polynomial $\hat{E}_{k,n}(q)$ is a $q$-analogue of the Eulerian numbers and interpolates between Eulerian, Narayana, and binomial numbers at $q=1$, $0$, and $-1$. This formula was motivated by experimental evidence, while comparable expressions for other coefficients were not known in the source.