Universality conjecture for normalized collision times

Let $T$ be the total time considered above, with expectation and variance denoted by $\mathbb E[T]$ and $\operatorname{Var}(T)$, and let $N$ and $M$ be the parameters satisfying $N \gg M$. Let $\mu$ be a distribution in the stated wide class. Universality conjecture. The normalized variable has distribution function

$$\mathbb P \left( \frac{ T - \mathbb E[T]}{ \sqrt{\operatorname{Var}(T)}} \leq t \right) = 1 - e^{-t-1}.$$

This predicts a universal limiting law for the relevant stopping time when $N$ dominates $M$; the source does not provide a proof or a resolution, and the intended range of distributions $\mu$ should be checked.