Sharp linear critical-point bound for quadrature domains

Let $\Omega\subset\mathbb C$ be a quadrature domain of order $n$ and connectivity $k$, and let $v$ be its landscape function. Write $N$ for the number of critical points of $v$ in $\Omega$.

Quadrature-domain critical-point conjecture. The number of critical points satisfies

$$N\leq 2n-2+k.$$

This is the specific linear bound proposed after the paper's general conjecture that the critical-set size should grow linearly with the order and connectivity. The source provides no evidence that the bound has been proved or refuted.