Nikiforov's extremal p-energy conjecture for trees

Let $T=(V,E)$ be a tree with $|V|=n$ and $|E|=m$. Let $S_n$ be the star on $n$ vertices, let $P_n$ be the path on $n$ vertices, and let $\mathcal{E}_p(T)$ denote the $p$-energy of $T$.

Nikiforov's extremal $p$-energy conjecture. For $1\leq p\leq2$,

$$\mathcal{E}_p(S_n)\leq\mathcal{E}_p(T)\leq\mathcal{E}_p(P_n).$$

For $p>2$,

$$\mathcal{E}_p(P_n)\leq\mathcal{E}_p(T)\leq\mathcal{E}_p(S_n).$$

The first range has been proved, as has the upper bound in the second range. The lower bound for $p>2$ is known for positive even integers, while the full remaining assertion is open.