Unicyclic graph square-energy conjecture

Let $G$ be a unicyclic graph of order $n$, and let $k$ be the odd length of its unique cycle. Unicyclic graph square-energy conjecture.

$$\begin{cases} s^+(G)>n>s^-(G),&k\equiv3\pmod 4,\\ s^+(G)<n<s^-(G),&k\equiv1\pmod 4. \end{cases}$$

The conjecture is motivated by an exact calculation for cycles and further computational investigation. The source identifies unicyclic graphs as among the hardest cases for the main square-energy conjecture; no general proof is supplied.