Cao and Dehmer's monotonicity conjecture for degree-based graph entropy

Let $G$ be a connected graph, and let $I_{f^l_d}(G)$ denote its degree-based graph entropy as a function of the positive parameter $l$.

Cao and Dehmer's conjecture. For $l>0$, $I_{f^l_d}(G)$ is a monotonically increasing function of $l$ for connected graphs.

The conjecture is supported in the paper by numerical results for trees with $11$ vertices and connected graphs with $7$ vertices. Its validity for all connected graphs and all positive parameter values remains open.