Definition of ZLNHe:
1. A zero-information (0-info) election is an election about which all that is known is the candidates and the voting system. There's no information about the voters, their preferences, or any predictive information about details of the voting.
2. To vote a candidate at bottom is to not vote that candidate over anyone. To vote a candidate above bottom is to vote that candidate over someone.
Zero-Info LNHe (ZLNHe):
In a 0-info election, voting above bottom one or more of some certain set of candidates shouldn't decrease the probability that the winner will come from that set, as compared to voting them all at bottom.
[end of ZLNHe definition]
ZLNHe could be called a "weakening" of LNHe. But calling it "weak LNHe" would be misleading, because it is only very slightly weaker than LNHe.
It's easier to refer to a 0-info election than to try to name different kinds of voting-predictive information and stipulate them to be absent. But the information that actually must be absent in that criterion's scenario is information that is usually or always at least mostly absent even in non-0-info elections. Therefore, ZLNHe is nearly the same thing as LNHe, and the word "weakening" hardly even applies.
Definition of Strong ZLNHe:
Same as ZLNHe, except that voting one or more members of that set over bottom must increase the probability that the winner will come from that set.
[end of Strong ZLNHe definition]
Of course all methods that meet LNHe also meet ZLNHe.
Symmetrical ICT meets Strong ZLNHe.
Ordinary ICT, and traditional Condorcet methods don't comply with ZLNHe or Strong ZLNHe.