Veronese secant-index conjecture

Let $v_d^{(n)}(\mathbb{P}^n)$ be the degree-$d$ Veronese variety, let $\mathfrak{L}(X)$ be its sequence of secant indices, and let ${\mathfrak{R}\mathfrak{L}}^G(X)$ and $\mathfrak{R}\mathfrak{L}(X)$ be the greedy and reducible secant-index sequences defined in the source. Veronese secant-index conjecture. For each $n,d>1$, with $X=v_d^{(n)}(\mathbb{P}^n)$,

$${\mathfrak{R}\mathfrak{L}}^G(X)={\mathfrak{R}\mathfrak{L}}(X)=\mathfrak{L}(X).$$

Equivalently, the maximal number of points in the finite reduced intersection of $r$ linearly independent degree-$d$ hypersurfaces in $\mathbb{P}^n$ is ${\mathfrak{R}\mathfrak{L}}^G_{N_d-r-1}(X)$. The conjecture is presented as a quasi-enumerative analogue of Bézout's theorem and is supported, conditionally, by the Eisenbud–Green–Harris conjecture.