Asymptotic behavior of the digraph Poljak–Rödl function

Determine whether lim sup g(n)/n = 0 for the digraph Poljak–Rödl function g(n) = min{χ(G × H): G and H are digraphs with χ(G) = χ(H) = n}.

Background

The digraph analogue g(n) is less studied than the graph function f(n). Existing results give upper bounds such as g(n) ≤ 2n/3 and g(n) ≤ ⌈n/2⌉ + 3, but the paper identifies the asymptotic ratio as unresolved.