The Gompertz approximation conjecture for ties in the M&M Game

Let $P_{\text{tie}}(\lambda)$ denote the probability of a tie in the M\textup{&}M Game, where the probability of flipping heads evolves according to an exponential distribution, and let $\lambda$ be the rate parameter controlling how quickly that probability increases as M\textup{&}M's are depleted. For large $n$, the Gompertz approximation conjecture. the probability of a tie is primarily governed by $\lambda$ and satisfies

$$P_{\text{tie}}(\lambda)\approx\exp\left(-1.22\cdot\exp(-\lambda)\right).$$

This is an empirical approximation motivated by simulation results and a Gompertz curve fit; the supplied text gives no proof or resolution, so its status remains open.