Nikiforov's spectral Erdős–Sós conjecture for trees

Let $F$ be a tree of order $\ell\geq 4$. For integers $n,k,p$ with $n>k>0$ and $p\in\{0,\dots,\lfloor(n-k)/2\rfloor\}$, let $S_{n,k}^{p}$ be the graph obtained from $K_k\nabla(n-k)K_1$ by embedding $p$ independent edges into $(n-k)K_1$. For $n\geq\ell\geq 4$, define

$$G_{n,\ell}=S_{n,(\ell-2)/2}^{0}$$

if $\ell$ is even and $G_{n,\ell}=S_{n,(\ell-3)/2}^{1}$ otherwise. Nikiforov's conjecture. Let $\ell\geq 6$ and let $G$ be a graph of sufficiently large order $n$. If

$$\rho(G)\geq\rho(G_{n,\ell}),$$

then $G$ contains all trees of order $\ell$ unless $G=G_{n,\ell}$. This is a spectral analogue of the Erdős–Sós conjecture, which concerns forcing all trees of order $\ell$ from an average-degree condition. The conjecture is attributed to Nikiforov; its resolution status is not established by the supplied text.