Asymptotic chromatic number of products of tournaments

Determine the limit as n tends to infinity of the minimum χ(S × T) over all n-tournaments S and T.

Background

For products of n-tournaments, the paper reports that the minimum product chromatic number is asymptotically λn for some 1/2 ≤ λ ≤ 2/3. The exact asymptotic constant, or equivalently the stated limit, remains unresolved.

References

What is $\lim_{n \to \infty} \min { \chi (S \times T): \text{ $S$ and $T$ are $n$-tournaments} }$ ?

— A survey on Hedetniemi's conjecture  (2502.16078 - Zhu, 22 Feb 2025) in Section 5, Question q6