Enumeration of support tau-tilting objects for unary-binary arbors

Prove that the number of support tau-tilting objects associated with the gentle quiver-with-relations G_t constructed from a unary-binary arbor t equals the number of elements of the arbor poset P_t.

Background

For a unary-binary arbor, the paper constructs a gentle quiver-with-relations G_t by orienting the arbor and imposing specified zero-relations at binary vertices. The associated finite mutation graph is the Hasse diagram of a congruence-uniform poset, and its vertices are support tau-tilting objects. The conjecture proposes that this representation-theoretic enumeration agrees with the lattice-point enumeration of P_t. The equality is checked in the example presented and in small cases.