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The sevens team was scheduled to travel to the Gold Coast athletes' village on Wednesday, one day before the Commonwealth Games start, but with all players now needing to be assessed they will remain in isolation until full medical checks are carried out.
A joint statement from New Zealand Rugby and the NZ Olympic Committee said the team would stay in camp on the Sunshine Coast until Sunday, April 8. The women's team starts competition from April 13.
Mumps is a highly contagious virus, particularly in its early stages, and there will now be fears it will spread throughout the team.
The New Zealand team medical lead and team psychologist will travel to support the sevens side on the Sunshine Coast.
Should other team members be struck down, it has the potential to derail the women's campaign.
A mumps outbreak started in Auckland last year and it spread to the All Blacks ahead of their northern tour with Jack Goodhue, Rieko Ioane and Luke Romano all suffering from the illness.
Tui, an Olympic silver medalist, could be ruled out of the Games with usual recovery ranging anywhere from one week to a month.
"If you look at the timing around when you're supposed to be quarantined it's going to be a challenge," NZ chef de mission Rob Waddell told the New Zealand Herald. "We're making sure we understand the vaccinations in the team and seeing their wellbeing is top of the list.
"We're watching that situation really closely. We're concerned for the athlete in the first place and making sure she's well. They are isolated up there, and given we only got the diagnosis at 6am this morning we're working through it and making sure we're going to get the best outcome."
Tui's sudden sickness continues an eventful build-up for both the men's and women's teams, both of whom have already lost players to injuries.
Earlier on Tuesday it was announced Kat Whata-Simpkins suffered a leg injury and would miss the Games.
That saw Tenika Willison promoted into the squad of 12 and Alena Saili join the team as the travelling reserve.
News wasn't any better for the men's side after emergency appendix surgery took talented Blues teenager Caleb Clarke out of contention.
Clarke's illness came after young playmaker Ngarohi McGarvey-Black was forced to return home after injuring his ankle in a warm-up match against Australia.
Clarke, the 13th man for the Games, is understood to have not been able to eat all Monday and gone under the knife early Tuesday morning.
The timing of his illness could not be worse but at least the 19-year-old is likely to get another crack in future.
Etene Nanai-Seturo, left out of the initial squad, flew to the Gold Coast as a replacement for McGarvey-Black on Sunday.
A second replacement player for Clarke may now be needed.