Extension conjecture for holomorphic Engel plane fields

Let $M$ be a complex manifold and let $S \subset M$ be a subset of complex codimension at least two. A holomorphic Engel $2$-plane field is a holomorphic $2$-plane field satisfying the Engel bracket-generating condition. The Engel extension conjecture. Holomorphic Engel $2$-plane fields extend holomorphically across subsets of complex codimension at least two as holomorphic $2$-plane fields, and across subsets of complex codimension at least three as holomorphic Engel $2$-plane fields. Examples show that the Engel condition need not extend across codimension-two subsets, so the stronger codimension-three threshold is essential to the claim.