Extremal critical-point conjecture for quadrature domains

Let an RL-domain be a domain of order $n$ and connectivity $k$ whose landscape function is defined as in the paper. For integers $n\geq 2$ and $1\leq k\leq n+1$, the conjecture is that there exists an RL-domain $\Omega$ of order $n$ and connectivity $k$ whose landscape function has exactly

$$4n+k-6$$

critical points in $\Omega$.

Extremal critical-point conjecture. For every $n\geq 2$, and for each $k=1,2,\ldots,n+1$, such a domain exists. The bound is attained for $k=n+1$ by the extremal construction described in the paper; existence for all connectivities is supported by computer simulations but remains unproved.