Kruft's lower general position number conjecture for Cartesian products

Let $G$ and $H$ be graphs, let $G\mathbin{\square}H$ denote their Cartesian product, and let $\operatorname{gp}^-(G)$ denote the lower general position number of $G$. Kruft's conjecture. For any graphs $G$ and $H$,

$$\operatorname{gp}^-(G\mathbin{\square}H)\geq\min\{\operatorname{gp}^-(G),\operatorname{gp}^-(H)\}.$$

The inequality has been verified for all pairs of graphs of order at most six, but remains open in general.