Bipodal optimizers in the upper-tail variational problem for regular subgraph densities

Let $H$ be a fixed $d$-regular graph with $d\ge2$, and let $t(H,\cdot)$ denote its homomorphism density. We study the upper-tail event $t(H,G(n,p))\ge r^{|E(H)|}$ for fixed $0<p<r<1$ in a dense Erdős--Rényi random graph $G(n,p)$. Near the Lubetzky--Zhao replica-symmetric phase boundary and away from