The random-eigenvector probability conjecture for transitive graphs

Let $G$ be a transitive $d$-regular graph, let $\lambda$ be an eigenvalue of its adjacency matrix, and let $I_{+}$ be the independent set obtained from a random unit eigenvector in the $\lambda$-eigenspace by selecting vertices whose value is larger than at each neighbor. Write $q_d(\lambda)$ for the relative volume of the regular spherical simplex associated with pairwise angle $\arccos\left(\frac{d-2-\lambda}{2(d-1)}\right)$.

Random-eigenvector probability conjecture. For any transitive graph $G$,

$$P(v \in I_{+}) \geq q_d(\lambda)$$

for any eigenvalue $\lambda$, or at least for sufficiently small $\lambda$, namely $\lambda\leq\lambda_0$ for some $\lambda_0$. Consequently, if $\lambda_{\min}\leq\lambda_0$, the independence ratio of $G$ is at least $q_d(\lambda_{\min})$.

For cherry-transitive graphs equality holds by symmetry, so the conjecture asserts that the probability is no smaller in the general transitive case. Its status is not resolved in the supplied source.