Generalized clique-counting conjecture for trees

Determine whether, for every t-vertex tree T_t, integers α≥1, 0≤β≤t−2, 3≤r≤t−2, and n=α(t−1)+β, the generalized Turán number satisfies ex(n,K_r,T_t)=α\binom{t−1}{r}+\binom{β}{r}.

Background

The conjecture proposes that the disjoint union αK_{t−1}∪K_β maximizes the number of r-cliques in every n-vertex graph that contains no copy of a prescribed t-vertex tree. The paper proves the formula for trees with sufficiently large common leaf bunches and for the large-clique regime r=t−d under explicit parameter bounds, but does not establish it for all trees and all permitted clique orders.