Before revision:  In addition to its inherent combinatorial interest---for instance, the t=3 case is connected to Tuza's famous conjecture on covering triangles of a graph with edges---variants of AHTP arise in recently proposed reductions to fundamental Euclidean clustering problems.
After revision:  Aharoni and Zerbib also posed whether Tuza's famous conjecture on covering triangles of a graph with edges---variants of AHTP arise in recently proposed reductions to fundamental Euclidean clustering problems.