On Friday, the boys' mother and father, Stephanie Archer and Vaughn Snowling, were devastated when told by specialists from Starship Children's Hospital in Auckland that none of the family were suitable donors.
But tears soon turned to joy when informed the babies' blood results had been improving - and the leukemia could be gone before their first birthday.
"We were given heartbreaking then hopeful news in the space of a few minutes," Archer said.
"It is now thought likely the babies have an as-yet unidentified syndrome.
"It means, all going well, the cancer could go away. The boys' enlarged livers and spleens will revert back to normal and their marrow should eventually start functioning normally."
The babies will be seen by a geneticist this week to try to identify the syndrome for appropriate treatment.
However, both boys still face challenges.
Laken needs open heart surgery and Theo has two faults that are being monitored; they are still very small for their age and are struggling to put on weight.
"They are not out of the woods yet but there is now no rush for a bone marrow transplant," Archer said. "We have new hope and the future is looking much brighter.
"We will always help our little battlers with whatever life throws at them and the help and support we have had from friends, family, medical staff and wellwishers is unbelievable."