Boundary extremality conjecture for the normalized simplex of spectra

Let $\mathbb{SL}_1^n$ denote the normalized set of spectra under consideration, let $\partial \mathbb{SL}_1^n$ be its boundary, and let $\mathbb{E}^n$ be its set of extremal points. Boundary extremality conjecture. If $n>3$, then

$$\partial \mathbb{SL}_1^n\subseteq\mathbb{E}^n,$$

i.e., every point on the boundary is extremal. The conjecture would show that characterizing the extreme points is enough to characterize $\mathbb{SL}_1^n$; it is known to fail when $n=2$ and $n=3$, while its validity for $n>3$ remains open.