Close Divisors of Typical Integers:The Ford--Green--Koukoulopoulos Conjecture

For an integer $k\geq2$, let $α_k$ be the supremum of the real numbers $a$ for which almost every integer $n\geq2$ has divisors $d_1<\cdots<d_k\mid n$ satisfying $d_k\leq d_1\bigl(1+(\log n)^{-a}\bigr).$ Let $\mathcal A\subseteq\N$ be the logarithmic random set in which the events $m\in\mathcal A$ a