Common-boundary parameters lie on the principal Thurston-set component boundary

Let $\mathcal{M}$ be the connectivity locus and let $\mathcal{M}_0$ be the Thurston set, both contained in the unit disk $\mathbb{D}$. Let $U$ be the connected component of $\mathbb{D}\setminus\mathcal{M}_0$ containing $0$. Common-boundary conjecture. Every parameter

$$\lambda\in\partial\mathcal{M}\cap\partial\mathcal{M}_0$$

belongs to $\partial U$.

This is motivated by a corollary concerning shared boundary points and by experimental evidence. The supplied text does not give a proof or a resolution, so the assertion remains open.