Anderson–Klee's projective dual singular-locus conjecture

Let $X\subset\mathbb{P}^n$ be a smooth, irreducible, non-degenerate projective variety. For each $r\in\{0,\ldots,n\}$, define

$$X^*\langle r\rangle=\left\{H^{\bot}\in X^*: \dim H^0\left(\mathcal{J}_{p(q^{-1}(H^{\bot}))}(1)\right)\leq n-r\right\},$$

where $p(q^{-1}(H^{\bot}))$ is the contact locus of $H$ along $X$ and $\mathcal{J}_{p(q^{-1}(H^{\bot}))}$ is its ideal sheaf. Anderson–Klee's projective dual singular-locus conjecture. For all $r\in\{0,\ldots,n\}$,

$$\dim X^*\langle r\rangle\leq n-r-1.$$

This is the algebraic-geometric form of the Anderson–Klee principle, imposing a codimension bound on dual hyperplanes according to the linear conditions cut out by their contact loci. The source describes it as a conjecture in the smooth projective setting.