Parallel k-partition conjecture for incidence configurations

Let $(\mathcal P,\mathcal H)$ be a point-hyperplane configuration in $\mathbb R^d$. A parallel $k$-partition is a partition of $\mathcal H$ into blocks of at most $k$ parallel hyperplanes such that every point is incident to a hyperplane in each block.

Parallel $k$-partition conjecture. For every fixed integer $k>1$,

$$\operatorname{rs}(\mathcal P,\mathcal H) \geq mn\cdot 2^{-\operatorname{polylog}(d)}.$$

Equivalently, every $k$-listable matrix $M\in\mathbb R^{n\times m}$ contains a $1$-listable submatrix of size at least $mn\cdot 2^{-\operatorname{polylog}(\operatorname{rank}(M))}$. The source proves this conjecture equivalent to the parallel 2-partition conjecture and hence to the log-rank conjecture, so it remains open.