Existence of non-monomial involutive dual-unitary Yang–Baxter gates

Determine whether there exists an involutive dual-unitary Yang–Baxter gate that is not of monomial type, equivalently whether every involutive dual-unitary Yang–Baxter gate is monomial up to an overall phase and a homogeneous onsite change of basis.

Background

The paper proves polynomial operator-Schmidt-rank bounds for all involutive dual-unitary Yang–Baxter gates, while separately analyzing phase-dressed monomial constructions. No example is known that is simultaneously involutive, dual-unitary, and non-monomial. The authors also explain that a proposed broader classification of unitary Yang–Baxter operators would not by itself settle this specific existence question.