The major-index polynomial-degree conjecture

Let $S_n$ be the symmetric group, let $\operatorname{maj}(\pi)$ denote the major index of $\pi$, and let $M(n,k)$ be the number of permutations $w\in S_n$ with $\operatorname{inv}(w)=k$. For fixed $k>0$ and $n\geq k$, consider the expected value of the first letter conditioned on $\operatorname{maj}(\pi)=k$.

Major-index conjecture. This expected value is a rational function in $n$ that is $1/(k+1)$ times the quotient of a monic degree-$(k+1)$ polynomial by a monic degree-$k$ polynomial. In particular, the source gives

$$\mathbb{E}[\pi(1)\mid \pi\in S_n,\operatorname{maj}(\pi)=1]=\frac{1}{2}\mathopen{}\mathclose\bgroup\originalleft(\frac{n^2+n-2}{n-1}\aftergroup\egroup\originalright),$$ $$\mathbb{E}[\pi(1)\mid \pi\in S_n,\operatorname{maj}(\pi)=2]=\frac{1}{3}\mathopen{}\mathclose\bgroup\originalleft(\frac{n^3-n-6}{n^2-n-2}\aftergroup\egroup\originalright),$$ $$\mathbb{E}[\pi(1)\mid \pi\in S_n,\operatorname{maj}(\pi)=3]=\frac{1}{4}\mathopen{}\mathclose\bgroup\originalleft(\frac{n^4+6n^3-13n^2-18n}{n^3-7n}\aftergroup\egroup\originalright),$$ $$\mathbb{E}[\pi(1)\mid \pi\in S_n,\operatorname{maj}(\pi)=4]=\frac{1}{5}\mathopen{}\mathclose\bgroup\originalleft(\frac{n^5+20n^4-45n^3-80n^2-16n}{n^4+2n^3-13n^2-14n}\aftergroup\egroup\originalright).$$

The denominator is an integer multiple of $M(n,k)$, a degree-$k$ polynomial. This is presented as a conjectured pattern for the major index; the supplied text gives no resolution evidence, so it remains open.