Abuaf's dimension conjecture for higher-dimensional tangency loci

Let $X \subset \mathbb{P}^N$ be a smooth complex projective variety. Denote by $X^*_r$ the variety

$$X^*_r:= \{ H^{\perp} \in X^*, \, \textrm{such that} \, \dim \langle (H \cap X)_{\tan} \rangle \geq r \}.$$

Abuaf's dimension conjecture. One has

$$\dim X^*_r \leq N-r-1.$$

This conjecture proposes a complex-algebraic analogue of the Anderson–Klee bound for singular points of convex bodies, controlling the dimension of the locus of hyperplanes whose tangency locus spans dimension at least $r$.