Characterization conjecture for maximal monotone operators with unique representation

Let $X$ be a Banach space and let $A:X\rightrightarrows X^*$ be a maximal monotone operator admitting a unique representative function. The representative functions are convex lower semicontinuous functions whose restriction to the graph of $A$ agrees with the duality pairing.

Characterization conjecture. The operator $A$ must either be the subdifferential $\partial f$ of a proper convex lower semicontinuous function $f$ of the form described in the paper, or be a linear skew-symmetric operator.

This conjecture aims to characterize all maximal monotone operators admitting a unique representative function. The paper notes that, apart from subdifferentials covered by its results, the only known non-subdifferential examples are skew-symmetric linear operators; whether the stated dichotomy holds remains open.