The Glivenko–Cantelli conjecture for bivariate record breaks

For each $M\geq1$ and $k\geq0$, let $\widetilde I_{m,k}$ be the indicator that the $m$th generated record breaks precisely $k$ remaining records, and define

$$\tilde p_{M,k}:=M^{-1}\sum_{m=1}^M\widetilde I_{m,k}.$$

Glivenko–Cantelli conjecture. The empirical fractions satisfy

$$\sup_{k\geq0}\left|\tilde p_{M,k}-2^{-(k+1)}\right|\overset{\mathrm{a.s.}}{\longrightarrow}0$$

as $M\to\infty$. This is a uniform empirical version of convergence to the geometric distribution with parameter $1/2$. The paper states that proving it can be reduced to an asymptotic variance calculation, which is left for future research.