Establish convexity of the normalized largest signless p-Laplacian eigenvalue

Prove or disprove that the function $p\mapsto \lambda^{(p)}_{\max}(G)/2^p$ is convex for every graph $G$.

Background

The authors report that numerical calculations show the curve $\lambda^{(p)}_{\max}(G)/2^p$ to be convex for all graphs examined. They do not establish whether this observed property holds universally, leaving its validity for arbitrary graphs unresolved.