Pomerance's $e^{-\gamma}J_0(x)$ threshold conjecture

Let $a_1,a_2,\ldots$ be independent uniform random integers in $\{1,\ldots,x\}$, and let $T$ be the least $t$ such that some subsequence of $a_1,\ldots,a_t$ has product equal to a square. Let $\pi(y)$ count the primes up to $y$, let $\Psi(x,y)$ count the $y$-smooth integers up to $x$, choose $y_0=y_0(x)$ maximizing $\Psi(x,y)/y$, and define

$$J_0(x):=\frac{\pi(y_0)}{\Psi(x,y_0)}x.$$

Let $\gamma=0.577\ldots$ be the Euler–Mascheroni constant. Pomerance's threshold conjecture. For every $\epsilon>0$,

$$\operatorname{Prob}\bigl(T\in[(e^{-\gamma}-\epsilon)J_0(x),(e^{-\gamma}+\epsilon)J_0(x)]\bigr)=1-o(1)$$

as $x\to\infty$. Thus the conjectured sharp threshold is centered at $e^{-\gamma}J_0(x)$. The source presents this as the authors' belief and does not give a proof or resolution.