Igusa–Todorov's conjecture on tilted algebras and maximal green sequences

Let $\Lambda$ be a cluster-tilted algebra of finite type, and let $B$ be one of the associated tilted algebras. Order the $B$-modules from right to left in the Auslander–Reiten quiver of $\Lambda$. A maximal forward hom-orthogonal sequence is a sequence of Schurian $\Lambda$-modules $M_1,\ldots,M_m$ such that $\operatorname{Hom}_\Lambda(M_i,M_j)=0$ for all $i<j$ and no other module can be inserted while preserving this property. A maximal green sequence is a maximal sequence of green mutations, and its $c$-vectors are the corresponding dimension vectors.

Igusa–Todorov's conjecture. (a) The ordered $B$-modules form a maximal forward hom-orthogonal sequence of $\Lambda$-modules whose dimension vectors form the $c$-vectors of a maximal green sequence for $\Lambda$. (b) The longest maximal green sequence for $\Lambda$ is given in this way.

This conjecture proposes that an associated tilted algebra simultaneously supplies a maximal forward hom-orthogonal sequence and a longest maximal green sequence for every cluster-tilted algebra of finite type. The supplied text gives no resolution status.