On the Gruss Inequality for unital 2-positive linear maps

In a recent work, Moslehian and Rajic have shown that the Gruss inequality holds for unital n-positive linear maps $\varphi :\mathcal{A}\to B(H)$, where $\mathcal{A}$ is a unital C*-algebra and H is a Hilbert space, if $n \ge 3$\mathrm{.}Theyalsodemonstratethattheinequalityfailstohold,ingeneral,if$n = 1$andquestionwhethertheinequalityholdsif$n=2$. In this article, we provide an affirmative answer to this question.