Equality of the bistochastic and correlation-matrix parameters

Let $\lambda_n$ be the smallest nonnegative number such that

$$\lambda_n \mathcal{B}_n^{sym}+(1-\lambda_n)W_n\subseteq\mathcal{B}_n^{sym},$$

and let $\rho_n$ be the smallest nonnegative number such that

$$\rho_n\varepsilon_n(\mathbb{R})+(1-\rho_n)I_n\subseteq\mathcal{Q}_n.$$

Here $\mathcal{B}_n^{sym}$ is the set of $n$ by $n$ symmetric bistochastic matrices, $W_n$ is the relevant distinguished symmetric bistochastic matrix, $\varepsilon_n(\mathbb{R})$ is the convex set of real correlation matrices, and $\mathcal{Q}_n$ is the convex hull of the real rank-one correlation matrices. Equality conjecture.

$$\lambda_n=\rho_n\quad\text{for all }n\in\mathbb{N}.$$

The conjecture is motivated by the close relationship between self-dual doubly stochastic maps, symmetric bistochastic matrices, and real correlation matrices. The equality is known for $n\leq 4$, where both parameters have the values $\lambda_1=\lambda_2=1$ and $\lambda_3=\lambda_4=\frac{2}{3}$, but the general case remains open.