Tightness of the constant in the average-degree independent-transversal theorem

Determine whether the constant 4 in the sufficient condition that a multipartite graph with maximum block average degree at most D and local degree sufficiently small has a 4D-thick partition guaranteeing an independent transversal is tight.

Background

Lemma 4.2, attributed to Kang and Kelly, states that for every ε > 0, sufficiently small local degree relative to the maximum block average degree D, and a partition whose blocks have size at least (1 + ε)D, an independent transversal exists. The paper notes that the preceding result improves on the earlier warm-up theorem when s = 1 and identifies the constant 4 in the 4D-thick condition as the best currently known absolute constant in the cited result.

The authors explicitly leave unresolved whether this constant can be reduced, so the problem concerns establishing the optimal threshold for block thickness in the locally sparse, average-degree setting for independent transversals.