Directly Observing Momentum Transfer in Twin-Slit "Which-Way" Experiments

Is the destruction of interference by a which-way measurement due to a random momentum transfer $\wp\agt\hbar/s$, with $s$ the slit separation? The weak-valued probability distribution $P_{\rm wv}(\wp)$, which is {\em directly observable}, provides a subtle answer. $P_{\rm wv}(\wp)$ cannot have support on the interval $[-\hbar/s,\hbar/s]$. Nevertheless, its moments can be identically zero.