<p>We show that the quantum advantages arising from incompatibility of a set of measurements cannot be enhanced by subjecting them to a filter, namely, by combining them with a device that post-selects the incoming states on a fixed outcome of a stochastic transformation. This result holds for several measures of incompatibility, such as those based on robustness and convex weight. As an application, we show that, for Einstein-Podolsky-Rosen steering, our no-go results determine the maximum steerability attainable under the most general local filters, together with an explicit expression for the optimal filter. Moreover, our results generalize to nonphysical maps, i.e., positive but not completely positive linear maps.</p>