Bednarska-Bzdęga–Hefetz–Łuczak conjecture for Waiter–Client subgraph counts

Let $H$ be a graph with $e(H)\ge 1$, and let $m(H)$ denote its maximum density over nonempty subgraphs, namely

$$m(H)=\max\left\{\frac{e(H')}{v(H')}:H'\subseteq H,\ v(H')\ge 1\right\}.$$

Consider the $b$-Waiter–Client game on the edges of $K_n$, where Waiter offers $b$ previously unchosen edges in each round and Client chooses one, which is added to the resulting graph. Bednarska-Bzdęga–Hefetz–Łuczak's Waiter–Client conjecture. For every graph $H$ with $e(H)\ge 1$, there exist $c,\gamma>0$ such that, for $n$ sufficiently large and $b\le c n^{1/m(H)}$, Waiter can ensure that the resulting graph contains at least

$$\frac{\gamma n^{v(H)}}{b^{e(H)}}$$

copies of $H$. This conjecture asserts the direction complementary to the already proved regime $b\gg n^{1/m(H)}$, and formalizes the claim that the game follows the probabilistic intuition of a random graph with the same density.