The high-dimensional zero-breaking record fraction conjecture

Consider dimension $d\geq 2$. Let $f_{d,m}$ denote the fraction of the first $m$ records generated that break $0$ records.

Zero-breaking record fraction conjecture. There exist constants $p_d\in(0,1)$ such that, almost surely,

$$f_{d,m}\to p_d\quad\text{as }m\to\infty.$$

Moreover,

$$p_d\to 1\quad\text{as }d\to\infty.$$

The conjecture concerns the limiting proportion of records that do not break any existing records in the multivariate-records process. It is motivated by simulation data in higher dimensions; the existence of the limits and their convergence to $1$ as the dimension grows remain open in the supplied source.