Does violation of a Bell inequality always imply quantum advantage in a communication complexity problem?

Quantum correlations which violate a Bell inequality are presumed to power better-than-classical protocols for solving communication complexity problems (CCPs). How general is this statement? We show that violations of correlation-type Bell inequalities allow advantages in CCPs, when communication protocols are tailored to emulate the Bell no-signaling constraint (by not communicating measurement settings). Abandonment of this restriction on classical models allows us to disprove the main result of, inter alia, [Brukner et. al., Phys Rev. Lett. 89, 197901 (2002)]; we show that quantum correlations obtained from these communication strategies assisted by a small quantum violation of the CGLMP Bell inequalities do not imply advantages in any CCP in the input/output scenario considered in the reference. More generally, we show that there exists quantum correlations, with nontrivial local marginal probabilities, which violate the $I_{3322}$ Bell inequality, but do not enable a quantum advantange in any CCP, regardless of the communication strategy employed in the quantum protocol, for a scenario with a fixed number of inputs and outputs