The amended Bronowski conjecture on identifiability and tangential projections

Let $X\subset\mathbb{P}^{N}$ be an irreducible and non-degenerate variety with non-degenerate Gauss map. The variety $X$ is $h$-identifiable if a general fiber of its $(h+1)$-secant map consists of one point, and let

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

be a general $(h-1)$-tangential projection from the span of $h-1$ general tangent spaces. Amended Bronowski conjecture. $X$ is $h$-identifiable if and only if the general $(h-1)$-tangential projection $\tau_{h-1}^X$ is birational. This revised form, which assumes non-degeneracy of the Gauss map, amends Bronowski's original conjecture, which was proved false; its validity is the subject of the cited work.