Positive p-energy path-minimization conjecture for connected graphs

Let $p\geq2$, and let $G$ be a connected graph with $n$ vertices. Let $P_n$ be the path graph on $n$ vertices, and let $\mathcal{E}_p^+(G)$ denote the positive $p$-energy, the sum of the $p$th powers of the positive adjacency eigenvalues of $G$.

Positive $p$-energy path-minimization conjecture.

$$\mathcal{E}_p^+(G)\geq\mathcal{E}_p^+(P_n)=\sum_{k=1}^{\left\lfloor\frac{n+1}{2}\right\rfloor}2^p\cos^p\left(\frac{k\pi}{n+1}\right).$$

This generalizes the positive square-energy lower-bound conjecture to $p$-energy and is proposed as an open problem; the surrounding results provide partial evidence, including the even-integer cases for the corresponding total energy.