Colorful Tverberg conjecture for regression depth

Let $r,d$ be positive integers, and let $A_1,\dots,A_{d+1}$ be sets of $r$ hyperplanes each in $\mathbb{R}^d$. A partition of their union into $r$ sets $B_1,\dots,B_r$ is colorful Tverberg conjecture for regression depth. required such that

$$|A_i\cap B_j|=1$$

for every $i\in[d+1]$ and $j\in[r]$, together with a point $q$ having positive regression depth for each $B_j$. The conjecture would provide a colorful analogue of Tverberg's theorem for regression depth. However, this conjecture and its natural extensions to $\mathbb{R}^d$ have been disproved.