Quasi-star or quasi-clique conjecture for path supersaturation

Let $k$ and $\ell$ be positive integers with $k>\ell$. For an $n$-vertex graph containing at least $m$ copies of the path $P_k$, let $\operatorname{satex}(n,P_k:m,P_{\ell})$ denote the minimum number of copies of $P_{\ell}$. Path quasi-star or quasi-clique conjecture. The minimum is attained asymptotically by either a quasi-star or a quasi-clique. The source gives no resolution of this conjecture.