Chromatic numbers and products

Let f(n) be the smallest number so that there are two n chromatic graphs whose product has chromatic number f(n). Under the assumption that a certain sharper result than one obtained by Duffus et al. [On the chromatic number of the product of graphs, J. Graph Theory 9 (1985) 487-495], and Welzl [Symmetric graphs and interpretations, J. Combin. Theory, Sci. B 37 (1984) 235-244], holds we will prove that f(n)⩾n/2.