The vertical non-uniqueness conjecture for PC connectives outside PU

Let $\texttt{PC}$ and $\texttt{PU}$ be the classes of connectives defined in the paper. Let vertical interderivability mean that, for a connective $\texttt{\#}$, the corresponding labelled sequents involving $\texttt{\#}$ and its companion connective $\texttt{\natural}$ derive one another in the relevant dimension. A connective is vertically unique when it satisfies this interderivability condition.

Vertical non-uniqueness conjecture. For every connective $\texttt{\#}$, if $\texttt{\#}\in\texttt{PC}$ but $\texttt{\#}\notin\texttt{PU}$, then $\texttt{\#}$ is not vertically interderivable, and hence is non-unique according to Definition 16.

The conjecture would classify all connectives in $\texttt{PC}\setminus\texttt{PU}$ as non-unique under the paper's preferred Belnap-style notion. The source states that proof-search complexity prevents establishing it and leaves it for future research.