Summary: Let \(X_1, X_2, \dots, X_n\) be a sequence of coherent random variables, i.e., satisfying the equalities \[X_j=\mathbb{P}(A|\mathcal{G}_j), \qquad j= 1, 2, \dots, n,\] almost surely for some event \(A\). This paper contains the proof of the estimate \[\mathbb{P}\bigg(\max_{1\leq i<j\leq n} |X_i - X_j| \geq \delta\bigg) \leq \frac{n(1-\delta)}{2-\delta} \wedge 1, \] where \(\delta \in (\frac{1}{2}, 1]\) is a given parameter. The inequality is sharp: for any \(\delta\), the constant on the right cannot be replaced by any smaller number. The argument rests on several novel combinatorial and symmetrization arguments, combined with dynamic programming. Our result generalizes the two-variate inequality of K. Burdzy and S. Pal [Adv. Appl. Probab. 53, No. 1, 133–161 (2021; Zbl 1482.60023)] and in particular provides an alternative derivation.