No phase transition for the lower-tail variational problem of bipartite graphs

For $0\leq q\leq p\leq 1$, let $\mathsf{LT}_p(H,q)$ be the problem of minimizing $\mathbb{E}[I_p(W)]$ over graphons $W$ subject to $t(H,W)\leq q^{e(H)}$, where $I_p$ is the rate function and $t(H,W)$ is the homomorphism density. Bipartite lower-tail conjecture. If $H$ is bipartite, then the constant graphon $W\equiv q$ is always the unique minimizer of $\mathsf{LT}_p(H,q)$.

The conjecture holds for every bipartite graph $H$ satisfying Sidorenko's conjecture, but may hold more generally. Its status is not resolved by the supplied text.