The inversion-number conjecture for the expected first letter

Let $S_n$ be the symmetric group, let $\operatorname{inv}(\pi)$ denote the number of inversions of $\pi$, and let $M(n,k)$ be the number of permutations $w\in S_n$ such that $\operatorname{inv}(w)=k$. For fixed $k$ and $n>k$, consider the expected value of the first letter conditioned on having $k$ inversions.

Inversion-number conjecture. The expected value is the rational function

$$\mathbb{E}[\pi(1)\mid \pi\in S_n,\operatorname{inv}(\pi)=k]=\frac{M(n+1,k)}{M(n,k)}.$$

This is one of the proposed patterns for expected first letters conditioned on Mahonian statistics. The supplied text gives no evidence that the formula has been proved or disproved, so its resolution remains open.