Effective-bound conjecture for incidence strata

For fixed $k\geq 1$, let $N_k(P)$ be the minimum integer $n$ such that $\Delta^k_P$ embeds into $\mathbb{A}^n$, and let $N_k$ denote the upper bound in the displayed chain

$$\max \{N_k(P)\mid P \in \mathbb{N}^k\}\le \max \{N_k(P)\mid P \in (\mathbb{C}^k)^{\circ}\}\le N_k.$$

Effective-bound conjecture. These three quantities equal $2^k-1$ for $k\geq 1$. This conjecture gives the expected sharp value of the embedding dimension bound for the incidence strata, while the source does not provide a resolution of it.