The degeneracy-order conjecture for the online fixed graph process

Let $H$ be an arbitrary fixed graph, and let $d$ be the degeneracy of $H$. Write $\tau({\mathcal P}_H,n)$ for the first time at which the semi-random graph process contains a copy of $H$.

Degeneracy-order conjecture. With high probability,

$$\tau({\mathcal P}_H,n)=\Theta\left(n^{(d-1)/d}\right).$$

The paper proves the corresponding upper bound for every fixed graph and the matching lower bound when $H=K_{d+1}$. The conjecture asserts that the same order holds for every fixed graph $H$.