The finite-graph INT_f-minimizing partition conjecture

Let $f:[0,1]\rightarrow\mathbb R$ be continuous and strictly concave, let $\epsilon>0$, and let $G$ be a finite graph. A partition $\mathcal Q$ of $V(G)$ is $\mathrm{INT}_f$-minimizing with $M$ parts if it has $M$ parts and minimizes $\mathrm{INT}_f(G;\cdot)$ among all partitions of $V(G)$ with $M$ parts. A partition is weak $\epsilon$-regular if it satisfies the weak regularity condition with error parameter $\epsilon$.

Finite-graph $\mathrm{INT}_f$-minimizing partition conjecture. For every continuous and strictly concave function $f:[0,1]\rightarrow\mathbb R$ and every $\epsilon>0$, there exist numbers $M,n_0$ such that, for every graph $G$ of order at least $n_0$, every $\mathrm{INT}_f$-minimizing partition of $V(G)$ with $M$ parts is weak $\epsilon$-regular.

The conjecture proposes that minimizing the entropy-like functional over a sufficiently large fixed number of parts yields a weak regularity approximation for all sufficiently large finite graphs. The supplied text does not state whether this has been proved or disproved.