Uniform positivity and boundedness conjecture for the entropy-maximizing permuton density

Let the data be non-degenerate in the sense of Definition

. Let $h\in \mathfrak F^{\mathrm x,\mathrm y,I}$ be a maximizer of $\mathfrak E$, and set $g=-h_{xy}$ to be the density of the corresponding permuton. Let $\Omega^{\mathrm x,\mathrm y,I}$ denote the domain of this permuton. **Uniform bounds conjecture.** There exist constants $0<c<C$, depending on all the data, such that

c<g(x,y)<C

for almost all points $(x,y)\in\Omega^{\mathrm x,\mathrm y,I}$. Proposition

applies only where $g$ is positive; this conjecture asserts that, for non-degenerate data, the maximizing density is everywhere positive and uniformly bounded above and below almost everywhere in the domain. The claim is not resolved in the supplied text.