Modified Bronowski conjecture on identifiability and secant-variety degree

Let $X\subset\mathbb{P}^N$ be an irreducible and non-degenerate variety. Let $\mathbb{S}ec_h(X)$ denote the Zariski closure of the union of the $(h-1)$-planes spanned by $h$ points of $X$, and set

$$c_h=\operatorname{codim}_{\mathbb{P}^N}(\mathbb{S}ec_h(X)).$$

A variety of minimal degree has the smallest possible degree for its codimension.

Modified Bronowski conjecture. If a general $(h-1)$-tangential projection of $X$ is birational and its image is a variety of minimal degree, then $X$ is $h$-identifiable and

$$\deg\mathbb{S}ec_h(X)=\binom{h+c_h}{h}.$$

The degree in the conclusion is the smallest possible degree of the $h$-secant variety. The conjecture was proposed as a modification of Bronowski's conjecture and is implied by the original conjecture, but the paper gives counterexamples to both statements.