The converse from completely uniquely paired to semidistrim lattices

Let $L$ be a lattice. It is completely uniquely paired if there is a unique bijection $\lambda:L\to L$ such that, for every $x\in L$, $\lambda(x)$ is maximal among the elements $z\in L$ satisfying $\mathsf{Pop}^{\downarrow}_L(x)=x\wedge z$, and $\lambda^{-1}(x)$ is minimal among the elements $z\in L$ satisfying $\mathsf{Pop}^{\uparrow}_L(x)=x\vee z$. A lattice is semidistrim when it has the semidistrim property defined in the paper. The semidistrim pairing conjecture. Every semidistrim lattice is completely uniquely paired. This would show that the pairing structure known for semidistrim lattices extends to a unique global bijection, called rowmotion in this setting.