Strong positive square-energy conjecture for graphs with many edges

Let $G$ be a connected graph of order $n$ and size $m$. Strong positive square-energy conjecture. If $m\ge n+1$, then

$$s^+(G)\ge n.$$

This conjecture strengthens the positive-energy side of the preceding conjecture for graphs having at least two more edges than vertices. It was proposed after computational investigation; the source reports checks of the conjectures for graphs up to $9$ vertices, but no general proof is given.