Joint equidistribution conjecture for statistics on beta(0,1)-trees

Let $\operatorname{root}$, $\operatorname{rzero}$, $\operatorname{rmod}$, and $\operatorname{sub}$ be the statistics on beta$(0,1)$-trees defined in the paper. Two pairs of statistics are jointly equidistributed when they have the same joint distribution over beta$(0,1)$-trees.

Joint equidistribution conjecture. The pairs

$$(\operatorname{root},\operatorname{rzero})$$

and

$$(\operatorname{rmod},\operatorname{sub})$$

are jointly equidistributed on beta$(0,1)$-trees.

The claim extends the paper's established equidistribution results for other pairs of statistics. It is stated without a proof, and the supplied text gives no evidence that it has been resolved.