The polynomial complexity conjecture for graphon constraint sets

Let $F_1,F_2,\dots,F_n$ be finite graphs, let $t_1,t_2,\dots,t_m$ be real numbers in $[0,1]$, and let $\mathcal{S}$ be the set of graphons $W$ satisfying

$$t(F_i,W)=t_i\quad\text{for }1\leq i\leq n.$$

A graphon $W$ has polynomial $L_1$-complexity if there is a $d>0$ such that, for every $\varepsilon>0$, some stepfunction with $O(\varepsilon^{-d})$ steps satisfies $\lVert W-W'\rVert_1\leq\varepsilon$. Polynomial complexity conjecture. The set $\mathcal{S}$ is either empty or contains a graphon of polynomial $L_1$-complexity. This conjecture connects graphon complexity with extremal combinatorics: it asserts that any consistent finite collection of homomorphism-density constraints has a solution admitting polynomial-size stepfunction approximations. The claim is presented as a conjecture and is supported by examples, but no resolution is given here.