You and Liu's Laplacian spectral ratio conjecture for trees

Let $T$ be a tree with $n\geq 3$ vertices. The Laplacian spectral ratio is

$$R_L(G)=\frac{\mu_1(G)}{\mu_{n-1}(G)},$$

where $\mu_1(G)$ and $\mu_{n-1}(G)$ are the largest and second smallest Laplacian eigenvalues of a connected graph $G$. You and Liu's conjecture.

$$R_L(K_{1,\,n-1})\leq R_L(T)\leq R_L(P_n),$$

and the left equality holds if and only if $T=K_{1,\,n-1}$, while the right equality holds if and only if $T=P_n$. This conjecture proposes that among trees on $n$ vertices the star minimizes and the path maximizes the Laplacian spectral ratio; the source notes that conditions in terms of diameter and maximum degree had been established, but provides no resolution of the full conjecture.