Let $S$ be the cone of positive semidefinite symmetric real matrices of size $n\times n$. The cone $S$ spans a $d:=n(n+1)/2$ dimensional vector space.

Let $C\subset S$ be a subcone formed by intersecting $S$ with a plane $\Pi$. Suppose that the plane $\Pi$ is of high codimension, say $\dim \Pi = t d$ where $1/2 < t < 1$ .

If a matrix $A\in C$ spans an extremal ray of $C$, then ${\mathrm{rank}}\, A\le n-1$. This is because $A$ cannot be an interior point of $S$. Can this bound be improved in general?