Bronowski's conjecture on identifiability and tangential projections

Let $X\subset\mathbb{P}^{hn+h-1}$ be an irreducible and non-degenerate variety of dimension $n$. A variety is $h$-identifiable when a general point in 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.

Bronowski's conjecture. The variety $X$ is $h$-identifiable if and only if a general $(h-1)$-tangential projection

$$\tau_{h-1}^X:X\dasharrow\mathbb{P}^n$$

is birational.

The conjecture connects identifiability with unexpected birational maps to projective space. The implication from $h$-identifiability to birationality is known, and the full equivalence is known for curves, smooth surfaces, and linearly normal smooth threefolds when $h=2$; the general case is refuted by counterexamples.