The bipartite lower-tail conjecture in the sparse limit

Let $H$ be a graph and $0\leq r\leq 1$. Define $h(x)=x\log x-x+1$ and let $\mathsf{LT}(H,r)$ be the problem of minimizing $\mathbb{E}[h(W)]$ over graphons $W$ subject to $t(H,W)\leq r^{e(H)}$. Sparse bipartite conjecture. For every bipartite graph $H$ and every $0\leq r\leq 1$, the constant graphon $W\equiv r$ is the unique minimizer of $\mathsf{LT}(H,r)$.

This is the sparse-limit analogue of the finite-$p$ bipartite lower-tail conjecture. The supplied text does not state a resolution; the result is known in cases covered by Sidorenko's conjecture.