Can coarse measurements reveal macroscopic quantum effects?

It has recently been conjectured that detecting quantum effects such as superposition or entanglement for macroscopic systems always requires high measurement precision. Analyzing an apparent counter-example involving macroscopic coherent states and Kerr non-linearities, we find that while measurements with coarse outcomes can be sufficient, the phase control precision of the necessary non-linear operations has to increase with the size of the system. This suggests a refined conjecture that either the {\it outcome precision} or the {\it control precision} of the measurements has to increase with system size.