The vertical non-uniqueness conjecture for special cases

Let a special case be one of the trivially conservative cases considered in the paper, and let vertical uniqueness mean the paper's vertical interderivability condition for the corresponding connective. For a dimension $i\in\{1,2\}$, write $[i:\texttt{\#}]s$ and $[i:\texttt{\natural}]s$ for the relevant labelled sequents.

Special-case vertical non-uniqueness conjecture. Special cases are not vertically unique.

The source explains that proving this requires finding, for each special case, a sequent $s$ and a dimension in which the relevant sequents cannot be derived from one another. It gives no such general proof and leaves the matter to future research.