The conjecture on unboundedly many local maxima of inducibility profiles

Let $\mathcal{M}$ be the family of complete multipartite graphs, and let $I_H(x)$ denote the inducibility profile of a graph $H$. Unbounded-local-maxima conjecture. For every $k>0$, there is a graph $H\in\mathcal{M}$ such that $I_H(x)$ has at least $k$ local maxima.

The theorem preceding this conjecture establishes strict non-global local maxima for $K_t^-$ for the listed values of $t$, and the paper's first example gives a graph with at least two local maxima. The conjecture asks whether complete multipartite graphs can realize arbitrarily many local maxima.