The quasiconcavity conjecture for Burkholder functions at the zero matrix

Let $\mathbf E:\mathbb R^{n\times n}\rightarrow\mathbb R$ be a Burkholder function, including the functions defined from the Burkholder expressions in the source. A function is quasiconcave at $A$ when the quasiconcavity inequality is imposed at that matrix; the source notes that quasiconcavity at some matrix implies quasiconcavity at the zero matrix for the relevant homogeneous-at-infinity setting.

Zero-matrix quasiconcavity conjecture. Burkholder functions are quasiconcave at the zero matrix.

The conjecture is motivated by the failure of general rank-one concavity to imply quasiconcavity and by the Beurling function, which is quasiconcave at the origin under the stated norm condition but not quasiconcave away from the origin when $p\ne2$.