Chomsky 2001 proposes a notion of Minimality in which, basically, evaluated phase-by-phase, an XP can only block a probe-goal relation between a probe P and a goal YP if the chain-head of XP intervene (ie (i) XP is a suitable goal for P, and (ii) XP asymmetrically c-commands YP).
We present an argument from Japanese Object-Honorific Agreement to argue that this notion of Minimality cannot be correct, or that it can only be correct under very specific assumptions. Those assumptions are arguably conceptually problematic. Instead, we propose a notion of Minimality where XP blocks the probe-goal relation between P and YP whenever any occurrence of XP intervenes. This conclusion raises questions about previously established cases of (non-)intervention (eg Icelandic cases).