Uniqueness conjecture for cograph minimal $(\infty,k)$-polar obstructions

A cograph minimal $(\infty,k)$-polar obstruction is a cograph that is not $(\infty,k)$-polar but whose every proper induced subgraph is $(\infty,k)$-polar; its type $(c,i)$ records its number $c$ of connected components and its number $i$ of isolated vertices. For integers $k,c,i$ satisfying

$$1\leq i\leq c-2\leq k,$$

Uniqueness conjecture. There exists exactly one cograph minimal $(\infty,k)$-polar obstruction of type $(c,i)$. This extends the cases for which uniqueness is established and leaves the remaining admissible types to be characterized.