The fixed-break strong law conjecture

For each $M\geq1$ and $k\geq0$, let $\tilde p_{M,k}$ be the fraction of the first $M$ records that break precisely $k$ remaining records. Strong law conjecture. For every fixed $k\geq0$,

$$\tilde p_{M,k}\overset{\mathrm{a.s.}}{\longrightarrow}2^{-(k+1)}$$

as $M\to\infty$. This is the pointwise strong-law consequence of the uniform Glivenko–Cantelli conjecture, and the paper notes that conversely the pointwise statement also implies the uniform one.