Lifting the clique-counting conjecture from r to r+1

Establish that if the generalized clique-counting formula ex(n,K_r,T_t)=α\binom{t−1}{r}+\binom{β}{r} holds for a given clique order r and a t-vertex tree T_t, then it also holds for clique order r+1, with n=α(t−1)+β and 0≤β≤t−2.

Background

For stars, Gan, Loh, and Sudakov established an implication from the validity of the conjecture at r to its validity at r+1. The authors explicitly conjecture that the same implication holds for every tree, which would provide an inductive route toward the general clique-counting conjecture.