Guo's monotonicity conjecture for positive square energy under edge addition

Let $G$ be a graph, and let $uv$ be a non-edge of $G$. Write $G+uv$ for the graph obtained by adding the edge $uv$, and let $s^+(G)$ denote the sum of the squares of the positive eigenvalues of the adjacency matrix of $G$.

Guo's monotonicity conjecture. Adding an edge should not decrease positive square energy:

$$s^+(G+uv)\geq s^+(G).$$

This generalizes the corresponding monotonicity of the spectral radius. The conjecture is presented as an open problem and motivates the paper's study of positive $p$-energies; monotonicity is known to fail for $1\leq p<3$ in the broader $p$-energy setting.