Ramsey multiplicity of apices of trees

A graph $H$ is common if its Ramsey multiplicity, i.e., the minimum number of monochromatic copies of $H$ contained in any $2$-edge-coloring of $K_n$, is asymptotically the same as the number of monochromatic copies in the random $2$-edge-coloring of $K_n$. Erdős conjectured that every complete graph is common, which was disproved by Thomason in the 1980s. Till today, a classification of common graphs remains a widely open challenging problem. Grzesik, Lee, Lidický and Volec [Combin. Prob. Comput. 31 (2022), 907--923] conjectured that every $k$-apex of any connected Sidorenko graph is common. We prove for $k\le 5$ that the $k$-apex of any tree is common.