Gerbner–Palmer's generalized Turán conjecture for trees

Let $T$ be a tree on $k$ vertices and let $r\geq3$. Write

$$n=a(k-1)+b,\qquad 0\leq b<k-1.$$

For a graph $G$, let ${\mathcal N}(K_r,G)$ denote the number of copies of $K_r$ in $G$, and let ${\mathrm{ex}}(n,K_r,T)$ be the maximum of this number over all $n$-vertex $T$-free graphs. Gerbner–Palmer's conjecture.

$${\mathrm{ex}}(n,K_r,T)={\mathcal N}\bigl(K_r,aK_{k-1}\cup K_b\bigr)=a\binom{k-1}{r}+\binom{b}{r}.$$

The conjecture asserts that $aK_{k-1}\cup K_b$ maximizes the number of $r$-cliques among all $n$-vertex $T$-free graphs. The paper verifies it for $r=k-2$ and for $r=k-3\geq5$, but the general statement remains open in the supplied source.