Exact Clique Counts in Graphs with a Forbidden Tree

For a fixed tree $T$ on $t$ vertices, let $\ex(n,K_r,T)$ denote the maximum number of $r$-cliques in an $n$-vertex $T$-free graph. Write $n=α(t-1)+β$ with $0\leβ<t-1$. A conjecture proposed in extremal-graph form by Alon and Shikhelman~\cite{alon2016}, later stated explicitly as an exact clique-coun