Guiduli's spectral dense-neighborhood conjecture

Let $G$ be a graph on $n$ vertices, let $T_r(n)$ denote the Turán graph, let $N(v)$ denote the neighborhood of a vertex $v$, let $d(v)$ denote its degree, and let $bb(G)$ denote the spectral radius of $G$. Guiduli's spectral dense-neighborhood conjecture. If

$$\lambda(G)\geq\lambda(T_r(n)),$$

then either $G\cong T_r(n)$, or there is a vertex $v$ such that

$$\lambda(G[N(v)])\geq\lambda(T_{r-1}(d(v))).$$

Furthermore, if

$$\lambda(G)>\lambda(T_r(n)),$$

then the conclusion holds for any vertex $v$ having maximum weight given by a positive eigenvector for $\lambda(G)$. The conjecture is presented as a spectral analogue of the dense-neighborhood theorem of Bollobás and Thomason and Erdős and Sós, which proved the corresponding ordinary Turán statement. The supplied text does not state whether Guiduli's spectral conjecture has been resolved.