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A major report in late 2017 uncovered "unacceptably high" rates of dog euthanasia, as well as high numbers of injuries and dogs unaccounted for.
Robertson said recent events showed the sector still had "some way to go" to reach the appropriate safety standards as recommended in the report.
"I have informed Greyhound Racing NZ that I am not satisfied the recommendations are being implemented in a way that is improving animal welfare, and with their failure to provide sufficient information on changes they are making.
"It is the responsibility of the industry to hold itself accountable and ensure the best possible standards of welfare for greyhounds.
"Should the review show that progress has not been sufficient, a further fundamental look at the greyhound racing industry may be required," Robertson said.
In January, animal rights group SAFE labelled the Whanganui Greyhound Track a "death track" and called for it to be shut down after five dog deaths over summer.
Associate Agriculture Minister Meka Whaitiri said Greyhound Racing NZ had ignored her earlier suggestion that it resume "regular progress reporting" on how it was proceeding with the 2017 report's proposals.
"This review will now address these matters," Whaitiri said.
The review, which is due to be completed by August 1, will be led by the chairman of the Racing Integrity Establishment Board and former senior judge Sir Bruce Robertson.