The variance bound conjecture for fixed-break record counts

Let $K_i$ be the number of remaining records broken at time $i$, let $I_{i,k}:={\bf 1}(K_i=k)$, and define

$$R_{n,k}:=\sum_{i=1}^n I_{i,k},\qquad \sigma_{n,k}^2:=\operatorname{Var}R_{n,k}.$$

Variance bound conjecture. For each fixed $k\geq0$,

$$\sigma_{n,k}^2=O\left((\log n)^2\right).$$

The paper explains that this bound would yield the observations-time strong law and therefore the Glivenko–Cantelli conjecture; it is proposed because the analogous bound is known for the total record count, but the required calculation is left open.