Constant-community approximation conjecture for subgraph-counting functions

Let $f$ be a subgraph-counting function. A symmetric matrix $X$ is called a block matrix with $k$ communities if there are a symmetric matrix $P\in\mathbb{R}^{k\times k}$ and a partition of the indices into $k$ disjoint communities such that $X_{ij}=P_{\ell_1,\ell_2}$ whenever $i$ and $j$ belong to communities $\ell_1$ and $\ell_2$, respectively. The notation $\mathcal{X}_f$ denotes the class used in the paper for the matrices associated with $f$.

Constant-community approximation conjecture. There is a constant $c$ independent of $N$, but dependent on the weights $\beta_i$, such that every $X\in\mathcal{X}_f$ is $o(n)$-close to a block matrix with no more than $c$ communities.

This conjecture would strengthen the stated theorem, which gives approximation within $\delta n+5000C_{\boldsymbol{\beta}}^2n^{15/16}$ using a number of communities depending on $\delta$. The conjectured bound would use a constant number of communities and an error that is sublinear in $n$; the supplied text gives no resolution status.