Failure of bounded $\bar\partial$-solvability on pseudoconvex domains of finite type

We show that for every integer $n\ge3$ there is a bounded pseudoconvex domain in $\mathbb{C}^n$ with smooth real-analytic boundary whose maximum boundary D'Angelo $1$-type is $16$, carrying a $\bar\partial$-closed form continuous on the closure and smooth in the domain with no bounded distributional solution. On each fixed domain, we also construct a uniformly bounded sequence of smooth $\bar\partial$-closed data for which every bounded solution has norm at least $ck^{1/320}$ for all sufficiently large $k$. Each datum in this sequence and a corresponding solution extend smoothly to a neighborhood of the closure. The obstruction comes from a singular curve in two weak complex tangential directions, whose normalization admits holomorphic functions that do not extend to a neighborhood of the curve.