Persistence of global hypoellipticity failure for worm domains

Let $D$ be a worm domain whose $\overline\partial$-Neumann problem is not globally hypoelliptic. Let $A$ be its boundary annulus, let $V\subset\mathbb C^2$ be an arbitrarily small open neighborhood of $A$, and let $\widehat{D}\subset\mathbb C^2$ be another bounded smooth pseudoconvex domain such that

$$\widehat{D}\cap V=D\cap V$$

and such that every point of $\partial\widehat{D}\setminus V$ is of finite type. The persistence conjecture. The $\overline\partial$-Neumann problem on $\widehat{D}$ is not globally hypoelliptic. This conjecture asserts that failure of global hypoellipticity persists when the boundary away from the annulus is replaced by finite-type points; the source indicates that this question is open and that the relevant global hypoellipticity behavior is poorly understood.