The faithfulness conjecture for the triangular functor

Let $\Sigma$ be a connected oriented marked surface other than a sphere with $4$ punctures or a projective plane with $2$ punctures, and let ${\bf F}:{\bf TSurf}^t\to{\bf Grp}$ be the functor constructed from the groups $\mathbb T_\Delta$, mutations, and surface morphisms. Triangular-functor faithfulness conjecture. The restriction of ${\bf F}$ to ${\bf Tsurf}_\Sigma^t$ is faithful. The conjecture concerns whether this functor detects all morphisms in the tagged triangulation category for the stated class of surfaces; the two excluded surfaces are exceptional cases in the formulation.