Failure of bounded ∂-solvability on pseudoconvex domains of finite type

We show that for every integer n≥3 there is a bounded pseudoconvex domain in Cⁿ with smooth real-analytic boundary whose maximum boundary D'Angelo $1$-type is $16$, carrying a ∂-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 ∂-closed data for which every bounded solution has norm at least ck1/320 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.