Exactness of the second semidefinite lifting for the complex cut polytope

Let $\mathrm{CUT}^4_\infty$ denote the complex cut polytope and let $\mathbf{L}(\mathscr{B}_{1})$ denote its second semidefinite lifting. The preceding rank-three exception concerns whether these two sets can differ.

Exactness conjecture. The second semidefinite lifting is exact for $\mathrm{CUT}^4_\infty$, i.e.,

$$\mathbf{L}(\mathscr{B}_{1})=\mathrm{CUT}^4_\infty.$$

The authors state that they are unable to prove or disprove the existence of rank-three points witnessing strict containment; numerical tests motivate this conjecture, so its status is open.