Level set Crouzeix conjecture for finite Blaschke products

Let $\Theta$ and $B$ be finite Blaschke products with $\deg B<\deg\Theta$. Let $M_\Theta$ be the matrix associated with $\Theta$, and define the $1/2$-level set

$$\Omega^B_{1/2}=\{z\in\mathbb{C}:|B(z)|<1/2\}.$$

Level set Crouzeix conjecture. One has

$$W(M_\Theta)\not\subseteq\Omega^B_{1/2}.$$

This is a weak version of Crouzeix's conjecture for finite compressions of the shift. It is known for certain classes of pairs $(B,\Theta)$, but most cases remain open.