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In a statement, New Zealand Rugby (NZR) announced that it had asked the franchise to review their protocols for post-match activity.
Bridge and Mo'unga were accused of inappropriate behaviour during the Crusaders' tour of South Africa in May.
Mo'unga was accused of spitting beer at a woman in a Cape Town bar and inappropriately touching her.
In the statement, NZR Head of Rugby Nigel Cass said NZR's Independent Complaints Service Manager Steph Dyhrberg had assisted NZR in carrying out a thorough and comprehensive investigation into the allegations, which NZR took into account in reaching their decision.
"NZR's investigation has found that the allegations against George Bridge were not upheld," said Cass.
"With regard to Richie Mo'unga, the NZR investigation found that the allegations also could not be substantiated. Given the seriousness of these allegations, and the potential consequences, we needed to be confident that what was alleged, actually took place, and we weren't," said Cass
Cass said Mo'unga acknowledged he had been drinking on the night when the complainant approached him with an allegation to which he responded in a way that was poor, but which NZR accepted was out of character.
"The way he responded to the complainant was inconsistent with NZR values and expectations, and he has acknowledged this and subsequently apologised.
"We feel that he has learned some valuable lessons and we are satisfied that he will not put himself in the same position again in future. We have made our expectations around behaviour clear to him. He has been reminded of his obligation to be a role model for the sport at all times, especially when approached by members of the public - even in a social setting," Cass said.
NZR has asked the Crusaders to urgently review their protocols for team post-match activity.
Crusaders CEO Colin Mansbridge said: "We will learn from this experience and use it as an opportunity for improvement. For us, this is not just about two members of our team; we all need to take some responsibility for this and learn from it. That is why we are reviewing our policies around alcohol and social media use to ascertain if our existing policies are robust enough and make changes if needed. If we apply the same growth mindset off the field as we do on the field, we'll all be better people as a result."
Cass concluded: "This whole case serves as a reminder to all our teams, managers, coaches and players and staff that their behaviour has to be of the highest standard at all times and especially in their interaction with the public and that they uphold the rugby values of respect and responsibility at all times."