The conjectured quadratic eigenvector for the position matrix

Let $M$ be the position matrix of the one-shelf shuffling machine, with entries $M_{ij}=p_{i,j}$, where $p_{i,j}$ is the probability that the card initially in position $i$ ends in position $j$. For $n\geq 3$, define a vector $\vec{v}$ by

$$v_k=n^2-3nk+\frac{3k(k+1)}{2}-1.$$

The quadratic eigenvector conjecture. $M$ has an eigenvalue-eigenvector pair $(\frac{1}{4},\vec{v})$ with multiplicity at least $1$. Numerical computation suggests this eigenvector formula; the conjecture concerns the spectrum of the position matrix and remains unproved in the supplied text.