Spectral stability conjecture for simple random walks on complete graphs

For $n\ge 2$, let $P^s$ be the non-lazy simple random walk on the unmixed complete graph $K_{n+1}$, and let $DF(P^s)$ be the corresponding linearized curvature-flow matrix.

Complete-graph spectral conjecture. The simple random walk $P^s$ is asymptotically stable, and the eigenvalues of $DF(P^s)$ are

$$-\frac{n-1}{n}\quad\text{with multiplicity }{n\choose 2},\qquad -\frac{n+3}{n^2}\quad\text{with multiplicity }n,$$ $$-\frac{n+3}{n}\quad\text{with multiplicity }{n\choose 2}-1.$$

As $n\to\infty$, these eigenvalues tend respectively to $-1$, $0$ (with $-\frac{n+3}{n^2}\approx-\frac1n$), and $-1$. Numerical verification is reported for $n=4,5,6,7$, while the general assertion remains open.