Support tau-tilting enumeration for unary-binary arbors

Prove that, for every unary-binary arbor t, the number of support tau-tilting objects of the gentle quiver-with-relations G_t equals the number of elements of the arbor poset P_t.

Background

A unary-binary arbor has singleton vertex labels and at most two subtrees at each vertex. From such an arbor, the paper constructs a gentle quiver-with-relations G_t. The associated finite mutation graph is the Hasse diagram of a congruence-uniform partial order, while P_t is the coordinatewise poset attached directly to the arbor. The conjecture asserts equality of their cardinalities, motivated by matching computations in examples; it was checked by computer for arbors of size at most 8.