Exact value of k2 for packing (1^2,2^k)-colorings of planar graphs with maximum degree at most four

Determine the minimum positive integer k2 such that every planar graph with maximum degree at most four admits a packing (1^2,2^{k2})-coloring, that is, a partition of the vertex set into two independent sets and k2 pairwise 2-independent sets.

Background

The paper defines k2 as the least k such that every planar graph with maximum degree at most four is packing (1^2,2^k)-colorable. It proves an upper bound k2 ≤ 7 and provides a lower bound k2 ≥ 4 through an explicit example, but the exact value remains unknown.