The Kakeya problem for codimension-two planes in $\mathbb{R}^n$

We study the $(n-2)$-plane Kakeya problem in $\mathbb{R}^n$ ($n\geq4$) subject to convex Wolff axioms. In other words, we study finite families of $δ$-neighborhoods of $(n-2)$-plane segments ($δ$-flakes) in $\mathbb{R}^n$, with the property that not too many of them can be contained inside a common