Bronowski conjecture for varieties with non-degenerate Gauss map

Let $X\subset\mathbb{P}^{N}$ be an irreducible and non-degenerate variety with non-degenerate Gauss map. A variety is $h$-identifiable when a general point of its $h$-secant variety lies on a unique $(h-1)$-plane spanned by $h$ points of $X$, and a general $(h-1)$-tangential projection is the projection from the span of the tangent spaces at $h-1$ general points.

Non-degenerate Gauss-map conjecture. The variety $X$ is $h$-identifiable if and only if a general $(h-1)$-tangential projection

$$\tau_{h-1}^X:X\dasharrow X_{h-1}\subset\mathbb{P}^{N_{h-1}}$$

is birational.

This is proposed after the paper's counterexamples to the unrestricted Bronowski conjecture, since those examples have degenerate Gauss map. The source does not report a resolution of this strengthened statement.