Landman's duality defect conjecture

Let $X$ be a smooth nonlinear subvariety of $\mathbb{P}^N$. Its dual variety $X^\vee\subseteq \mathbb{P}^{N\vee}$ is the closure of the set of hyperplanes tangent to $X$, and define its duality defect by

$$\operatorname{def}(X)=N-1-\dim(X^\vee).$$

Landman's duality defect conjecture. If

$$\dim(X)>\frac{2N}{3},$$

then $X^\vee$ is a hypersurface, equivalently $\operatorname{def}(X)=0$.

The conjecture was posed by Alan Landman in 1974. It is known in codimension $2$, and in codimension $3$ when $N$ is odd, but remains open in general, including codimension $2$ for Hartshorne's related complete-intersection conjecture.