The conjectured nonzero spectrum of the position matrix

Let $M$ be the position matrix of the one-shelf shuffling machine. Its eigenvalues describe the associated Markov chain on card positions; the previously established eigenvalues include $1$ and $0$ with the stated multiplicities. The nonzero-spectrum conjecture. The nonzero eigenvalues of $M$ are

$$\frac{1}{2^{2i}},\qquad i=0,1,\ldots,\left\lceil\frac{n-1}{2}\right\rceil.$$

Numerical evidence is said to strongly suggest this form, but no proof or resolution is supplied in the text.