Conjecture on minimum-degree construction time in the semi-random graph process

Let $n$ be the number of vertices, let $k=k(n)$, and let $\tau(\sigma,\operatorname{\mathcal{P}}_{\min,k})$ denote the first round in which Builder's graph has minimum degree at least $k$. The notation $k=o(n)$ means that $k/n\to 0$ as $n\to\infty$.

Minimum-degree construction-time conjecture. The conclusion of Theorem~~ holds for $k=o(n)$.

The preceding theorem proves the corresponding upper bound when $k=o(n^{1/2})$, while a weaker upper bound is established for $k=o(n)$. Thus the conjecture asserts that the sharper estimate from the second part extends to the larger range.