Spectral instability conjecture for simple random walks on hypercubes

For $d\ge 2$, let $P^s$ be the non-lazy simple random walk on the hypercube $Q^d=(K_2)^d$, and let the corresponding linearized curvature-flow matrix have largest eigenvalue $\lambda_{\max}$.

Hypercube spectral conjecture. The simple random walk $P^s$ is unstable, all eigenvalues of the corresponding linearized curvature-flow matrix are real, and

$$\lambda_{\max}=\frac{4}{d}.$$

The conjecture was verified numerically for $d=2,3,\dots,9$. The positive largest eigenvalue explains the asserted instability, but the general statement remains open.