Circular-domain critical-point conjecture

A circular domain is a domain whose boundary components are circles. Let $\Omega$ be a $k$-connected circular domain, let $v$ be its landscape function, and let $N$ be the number of critical points of $v$.

Circular-domain critical-point conjecture. For $k\geq 3$,

$$N\leq 5(k-2).$$

The conjecture is motivated by the corresponding extremal behavior for RL-domains. The source gives no resolution of this bound.