Li and Rousseau's asymptotic Ramsey conjecture for complete fans

Let $F_k=K_1+kK_2$ be the graph consisting of $k$ triangles sharing one vertex, and let $r(G,H)$ denote the Ramsey number of graphs $G$ and $H$. Here $k$ is a fixed positive integer and $n$ tends to infinity.

Li and Rousseau's conjecture.

$$r(K_n,F_k)=(1+o(1))\frac{n^2}{\log n}.$$

This conjecture extends the asymptotic prediction for $r(K_n,F_1)=r(K_n,K_3)$ and remains open. Known upper bounds and the classical lower-bound framework establish the scale of the quantity, but not the asserted asymptotic constant.