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Abercrombie's wife Monique and their three young children were granted a medical exemption from hotel isolation three days after arriving home from Australia.
The Breakers have based themselves across the Tasman to compete in the National Basketball League. They originally went to Melbourne but after a local outbreak of Covid-19, the team moved on to Hobart.
Following the move, Monique decided she and the children would return to New Zealand.
Before they left Australia, they applied for a medical exemption to isolate at home because two of the children have special needs.
They managed to obtain managed isolation (MIQ) vouchers the night before their flights.
On day one of isolation, Monique complained on social media about her hotel room being mouldy and about the time it was taking to process their medical exemption. In a tweet, she tagged Prime Minister Jacinda Ardern.
"We have two kids who have special needs and I don't really want to go into detail of what those are, but we had supporting evidence from GPs and other specialists to provide evidence on why it wouldn't be safe for them … to isolate in a facility," Abercrombie told RNZ's Checkpoint programme today.
"We sent that application off, and at the end of the day when the opportunity came up to grab a voucher and get home without having the results of that [application] we just had to take that opportunity when we could, and wait for that process to finish once they go back to New Zealand."
Two of his children are on the autism spectrum and have complex behavioural and medical needs, he said.
"It's obviously been deemed significant enough to be granted an exemption, which I hope people can understand."
Abercrombie said it might be difficult to understand why children with such needs would travel to Australia under Covid-19 restrictions, but he hoped people with families would understand as a father he wanted to be with his family for the six months the Breakers were booked to be there.
"When we came over, our situation was very different to what it is now. We were planning on being based in Melbourne and travelling from there for the duration of our season, and that situation changed very quickly."
Monique had made clear the details of their children's medical needs throughout the process of landing back in Auckland and arriving at the facility, he said.
"We applied for the [managed isolation] voucher at about 10pm on a Wednesday, we spent the next two hours packing, had one hour's sleep, and then jumped in the car to go to the airport the next day.
"We made every effort when they landed to make those facts known, but also arrived on the knowledge that if the exemption didn't come through then they would have to isolate at the facility. That's something they were prepared to do."
Monique had complained on Twitter about the conditions of their managed isolation room.
"In hindsight, making those thoughts public on social media probably wasn't the right way to go about things, but she was travelling on about two hours' sleep with three children.
"So when she arrived and there was mould inside the window sills of the room, in a room that was supposed to be completely cleaned in between guests to get rid of any viruses and germs - that was quite a shock to her.
"But the hotel rectified those as soon as they could."
He said it was not commenting on social media that helped his family obtain an exemption to isolate at home.
"We applied, like anyone else would, through the appropriate channels for a medical exemption, and were granted them based completely on medical grounds, irrelevant to the fact of who I am or who she was."
Abercrombie's situation came after news of a dying New Zealander who was initially denied a place in managed isolation as he attempted to return from overseas to say goodbye to his family.
Abercrombie said he could not comment on whether he was the recipient of special treatment.
"We're very grateful and thankful that we were able to get an exemption. We realise it's not something that's given to everyone, it's very hard to get and we're very lucky that we were able to get one."
'All applications are treated fairly' - MIQ
Checkpoint asked managed isolation management if the Abercrombie family received special treatment at any part of the process in applying for MIQ places or exemptions.
In a statement, it said: "All exemption applications are assessed on a case by case basis, against a set of strict criteria, and all applications are treated fairly.
"Any exemption application for medical reasons is assessed by public health authorities with the final decision made by MBIE MIQ.
"Because the purpose of managed isolation is to keep our communities safe from Covid-19, and given the public health risk, there is an extremely high threshold for approving applications. As such, the majority of applications are declined."