EISPN conjectures for Cartesian products of paths

Let $P_m$ denote the path of order $m$, and let $\Box$ denote the Cartesian graph product. For a graph $G$, let $EISPN(G)$ denote its efficient isolated private-neighborhood number.

EISPN path-product conjectures. If $k \ge 1$, then

$$EISPN(P_3 \mathbin{\Box} P_{2k})=6k.$$

If $m \ge 2$, then

$$EISPN(P_4 \mathbin{\Box} P_m)=4m.$$

These conjectures are posed from data for graph classes. The source gives no resolution; the first EISPN assertion is presented as an additional conjecture following the EIPN grid conjecture.