Construct a return statistic for general composition runs

Associate to every composition $c$ satisfying $c\le_c vc(B)$, with $c\neq(1,\dots,1)$, a statistic $return_c$ on the set $P_B$ of lattice paths bounded by a boundary path $B$ such that the joint distribution of $run_c$ and $return_c$ is symmetric on $P_B$.

Background

The paper defines composition runs $run_c$ for compositions componentwise bounded by the vertical composition type $vc(B)$ of a boundary path $B$. It proves that $run_c$ is jointly symmetric with the existing boundary return statistic $return$ on the boundary-constrained subset $CP_B$, but observes that the analogous symmetry with block return does not persist for general composition runs. The authors therefore ask whether a new statistic depending on $c$ can be defined on all bounded paths $P_B$ to restore a symmetry analogous to the classical $(run,return)$ symmetry.