Positive square-energy bounds for maximal planar and outerplanar graphs

A graph is maximal planar if it is planar and no edge can be added while preserving planarity, and maximal outerplanar if it is outerplanar and no edge can be added while preserving outerplanarity. Maximal planar and outerplanar square-energy conjecture.

$$\begin{aligned} &\text{If $G$ is maximal planar of order $n\ge10$, then }s^+(G)\ge3n;\\ &\text{if $G$ is maximal outerplanar of order $n\ge8$, then }s^+(G)\ge2n. \end{aligned}$$

These bounds are proposed as more accessible refinements of previously discussed questions for maximal planar graphs. They are motivated by computational investigation and super-additivity of square energy; the source gives no general proof.