Path energy monotonicity conjecture for unicyclic graphs

Let $G$ be a unicyclic graph of order $n$ whose cycle has size $k$. The path energy $PE(G)$ is defined for such graphs. Path energy conjecture for unicyclic graphs. $PE(G)$ depends only on the parameters $n$ and $k$. For fixed $n$, $PE(G)$ is a monotonically increasing function of $k$. The surrounding text states that the paper gives results resolving Conjectures 2, 3, and 4 from the cited work; this claim is one of those results and is therefore presented as solved.