X and Y are weakly incommensurable iff there is a dimension of evaluation where X beats Y and a dimension of evaluation where Y beats X. X and Y are strongly incommensurable iff they are weakly incommensurable and a rational agent doesn't have on balance reason to choose X over Y and doesn't have on balance reason to choose Y over X.
Weak incommensurability is precisely what is needed for the possibility of a rational agent choosing either over the other.
Weak incommensurability is evidence of strong incommensurability. But there are cases where weak incommensurability fails to yield strong incommensurability. One kind of case is extremity. If one is choosing between being a superb nurse and a very mediocre mathematician, there is weak incommensurability, but one may have on balance reason to be a nurse (all other things being equal). But when choosing between being a nurse and a mathematician and one's professional quality would be moderately close, it's also strongly incommensurable.