The variance bound conjecture for records without coordinate records

Let $R^{(0)}_{n,k}$ count, through time $n$, records that break precisely $k$ remaining records and do not set a record in either coordinate. Reduced variance bound conjecture. For each fixed $k\geq0$,

$$\operatorname{Var}R^{(0)}_{n,k}=O\left((\log n)^2\right).$$

The paper presents this as a simpler conjecture whose proof would reduce the corresponding variance problem for the full count $R_{n,k}$, after controlling the other coordinate-record contributions.