Clay's position-matrix eigenvalue conjectures for the single-shelf shuffle

Let $n\geq 3$, and let $M$ be the position matrix of a single-shelf shuffle, with

$$M(i,j)=\frac{1}{2^i}\left(\binom{i-1}{j-1}+\binom{i-1}{n-j}\right),\quad 1\leq i,j\leq n.$$

For a vector $v\in\mathbb{R}^n$, write $v(k)$ for its $k$th coordinate.

Clay's position-matrix eigenvalue conjectures. The following assertions hold:

1. $(1/4,v)$ is an eigenvalue-eigenvector pair of $M$, with multiplicity at least one, where

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

2. The non-zero eigenvalues of $M$ are $2^{-i}$, where $0\leq i\leq n-1$ is an even integer.

These conjectures concern the spectrum of the position matrix arising from the single-shelf shuffling model. The source attributes them to Clay and provides no evidence here that either conjecture has been resolved.