Besomi–Pavez-Signé–Stein second-neighbourhood conjecture for tree embeddings

Let $k$ be a positive integer. For a graph $G$ and a vertex $x\in V(G)$, let $N_G(x)$ be the neighbourhood of $x$, and let $N_G^2(x)$ be the set of vertices other than $x$ sharing a common neighbour with $x$. Besomi–Pavez-Signé–Stein second-neighbourhood conjecture. If

$$\delta(G)\geqslant k/2$$

and $G$ contains a vertex $x$ such that

$$\min\{|N_G(x)|,|N_G^2(x)|\}\geqslant 4k/3,$$

then $G$ contains a copy of every tree with $k$ edges. This replaces a maximum-degree hypothesis by a local first- and second-neighbourhood condition; no resolution is supplied in the source, so the conjecture remains open.