Exact value of k1 for packing (1,2^k)-colorings of planar graphs with maximum degree at most four

Determine the minimum positive integer k1 such that every planar graph with maximum degree at most four admits a packing (1,2^{k1})-coloring, that is, a partition of the vertex set into one independent set and k1 pairwise 2-independent sets.

Background

The paper introduces k1 as the smallest k such that every planar graph with maximum degree at most four is packing (1,2^k)-colorable. It proves an upper bound k1 ≤ 10 and gives a lower bound k1 ≥ 6 via a concrete example, leaving the exact value of k1 unresolved.