For a locally compact non-compact group \({\mathcal G}\), consider \(L_{\infty}(\mathcal G)\) as a left module over the algebra \(L_1(\mathcal G)^{\ast\ast}\) endowed with the first Arens product. The main purpose of the article under review is to solve a conjecture made by \textit{H. Hofmeier} and \textit{G. Wittstock} [Math. Ann. 308, 141-154 (1997; Zbl 0943.46042)] concerning the automatic boundedness of the corresponding module homomorphisms on \(L_{\infty}(\mathcal G)\), which is done by showing an even stronger result. Moreover, the author shows that the theorem of \textit{F. Ghahramani} and \textit{J. P. McClure} [Can. Math. Bull. 35, 180--185 (1992; Zbl 0789.43001)] on the automatic \(w^{\ast}\)-continuity of bounded \(L_{\infty}(\mathcal G)^{\ast}\)-module homomorphisms on \(L_{\infty}(\mathcal G)\) can be sharpened in a similar fashion. Namely, he proves that for a linear operator \(\Phi\) on \(L_{\infty}(\mathcal G)\) in order to be bounded and even \(w^{\ast}\)-continuous, it suffices that \(\Phi\) commutes with the module action of a small set of functionals consisting of \(w^{\ast}\)-limits of point evaluations.