Extremal signed Wiener index conjecture for trees

Let $(T,\sigma)$ be a signed tree on $n$ vertices. For the path $P_n$, let $+$ denote the constant signing that assigns $+1$ to every edge, and let $\alpha$ denote the alternating signing that assigns the first edge $+1$, the second $-1$, the third $+1$, and so on. Then the signed Wiener index tree conjecture.

$$W_\alpha(P_n)\le W_\sigma(T)\le W_+(P_n).$$

The upper bound follows from the corresponding classical result because $W_+(P_n)=W(P_n)$; the lower bound remains to be proved.