Conjecture on the k-general position number of infinite grids

Let $P^{\Box,n}_\infty$ be the $n$-dimensional infinite grid, and let $\mathrm{gp}^k(G)$ denote its $k$-general position number. Infinite-grid conjecture.

$$\mathrm{gp}^k(P^{\Box,n}_\infty)=(k-1)^{2^{n-1}}.$$

This conjecture is motivated by the preceding results on finite grids and by the known value for the relevant infinite-grid case. Its status and the range of parameters for which it remains open are not specified in the source.