Rarity of solutions to the complex Monge--Ampère equation on weakly pseudoconvex domains

We show that on any weakly pseudoconvex $B$-regular domain, the classical Dirichlet problem for the complex Monge–Ampère equation with ${\mathcal{C}}^{\mathrm{\infty }}$-smooth data does not in general admit ${\mathcal{C}}^{1,1}$-smooth solutions. This working draft is a prelude to potential-theoretic solutions to some extension problems for mappings that were thought to rely on such ${\mathcal{C}}^{1,1}$-smooth solutions.