Tverberg–Vrećica conjecture for regression depth

Let $1\leq k\leq d$ be integers, and let $A_1,\dots,A_{d-k+1}$ be finite arrangements of hyperplanes in $\mathbb{R}^d$. Assume that

$$|A_i|=(k+1)(r_i-1)+1$$

for positive integers $r_i$, for each $i=1,\dots,d-k+1$. A $k$-dimensional subspace $L$, a point $q\in L$, and a partition of each $A_i$ into $r_i$ parts $A_i^{(1)},\dots,A_i^{(r_i)}$ should exist such that

$$RD(A_i^{(j)},q,L)\geq 1$$

for every $i=1,\dots,d-k+1$ and $j=1,\dots,r_i$. This is a regression-depth generalization of the center transversal and Tverberg theorems. The corresponding question remains open for finite families of points and was conjectured by Tverberg and Vrećica in 1993.