Three symmetry conjectures for statistics on di-sk trees

Let di-sk trees be the combinatorial objects considered in the paper, and let $\mathsf{omi}$, $\mathsf{riop}$, $\mathsf{rpop}$, $\mathsf{iop}$, and $\mathsf{pop}$ denote the statistics on them. For statistics $A_1,\ldots,A_k$, write $(A_1,\ldots,A_k)\sim(B_1,\ldots,B_k)$ when the corresponding statistic tuples are equidistributed over di-sk trees. The statistics $\mathsf{riop}$, $\mathsf{iop}$, $\mathsf{top}$, $\mathsf{pop}$, and $\mathsf{rpop}$ form an equidistributed quintuple.

Three symmetry conjectures for di-sk trees. Over di-sk trees, the following symmetries should hold:

$$(\mathsf{omi},\mathsf{riop},\mathsf{rpop})\sim(\mathsf{omi},\mathsf{rpop},\mathsf{riop}),$$ $$(\mathsf{omi},\mathsf{iop},\mathsf{rpop})\sim(\mathsf{omi},\mathsf{rpop},\mathsf{iop}),$$ $$(\mathsf{riop},\mathsf{pop})\sim(\mathsf{pop},\mathsf{riop}).$$

These conjectures propose the remaining symmetries suggested by computations after several related symmetries over di-sk trees had already been proved. The source gives no resolution of these three claims, so their status remains open.