Quantum measurement incompatibility does not imply Bell nonlocality

We discuss the connection between the incompatibility of quantum measurements, as captured by the notion of joint measurability, and the violation of Bell inequalities. Specifically, we present explicitly a given a set of non jointly measurable POVMs $\mathcal{M}_A$ with the following property. Considering a bipartite Bell test where Alice uses $\mathcal{M}_A$, then for any possible shared entangled state $\rho$ and any set of (possibly infinitely many) POVMs $\mathcal{N}_B$ performed by Bob, the resulting statistics admits a local model, and can thus never violate any Bell inequality. This shows that quantum measurement incompatibility does not imply Bell nonlocality in general.