Excess bound conjecture for the positive semidefinite cone

Let $\operatorname{Sym}^k_+$ denote the cone of positive semidefinite matrices, and let $e(C)$ be the excess of a closed convex cone $C$ over the Lorentz cone. Positive-semidefinite-cone excess conjecture. One has

$$e(\operatorname{Sym}^k_+)<2.$$

The source cites Ambühl for motivation and experiments. If true, this would give dimension-independent probabilistic condition-number bounds for semidefinite programming; the source does not report a resolution.