DeMarco–Kahn upper tail conjecture for strictly balanced graphs and dense regimes

Let $H$ be a graph, let $m_H$ denote its appearance-threshold parameter, and let the DeMarco–Kahn upper tail conjecture be the assertion that the upper-tail estimate stated above holds. A graph is strictly balanced when the relevant density parameter is strictly maximized by $H$ rather than by a proper subgraph. DeMarco–Kahn's restricted upper tail conjecture. The DeMarco–Kahn upper tail conjecture is true for every strictly balanced graph $H$. Moreover, for every fixed $\gamma>0$, it is true under the additional assumption

$$p\geqslant n^{-1/m_H+\gamma}.$$

These are proposed restricted regimes in which the false general conjecture may still hold. The paper presents them as open problems rather than proving them.