All $d\otimes d$ dimensional entangled states are useful for the antidiscrimination of quantum measurements when $d$ is even

Piani and Watrous [Phys. Rev. Lett.102, 250501 (2009)] proved that all entangled states are useful for discrimination of quantum channels. We pose the same question in the context of antidiscrimination of quantum channels. We partially answer this by showing that for every $d\otimes d$ entangled state (with even $d$), there exist three projective measurements which are antidiscriminable (but not discriminable) with that input state but those three measurements are not antidiscriminable with the product probe.