Degeneracy conjecture for isolated complex tangents of 3-manifold embeddings

Let $M$ be a smooth 3-manifold smoothly embedded in $\mathbb{C}^3$, and let $p\in M$ be an isolated complex tangent. The degeneracy conjecture asserts that the Bishop invariant at $p$ must be degenerate.

The example preceding the conjecture gives an embedding of $S^3$ with exactly one complex tangent, whose Bishop invariant is degenerate. The authors report that they were unsuccessful in proving the assertion for a general isolated complex tangent, so the conjecture remains open.