The infant underwent his third round of surgery on September 3 following the "cowardly" and random attack in a Brisbane park.
Police have been searching for the man since he poured a thermos of hot coffee on the boy at Hanlon Park in Stones Corner on August 27.
The baby suffered significant burns to his face, arms and legs.
"Detectives are continuing to investigate the matter and are working with interstate and international partner agencies to progress the investigation," a police spokesperson said.
CCTV and enhanced images have been released but no arrests have been made.
Police have been unable to locate the man since the attack. (HANDOUT/QUEENSLAND POLICE SERVICE)
Police are expected to provide an update on the investigation on Monday.
The family said the little boy had been incredibly resilient despite the "traumatic and difficult" time.
"He's up and about, walking and causing chaos," his mother said.
"He is such a strong little boy and we are so so proud of him."
But the family is still seeking justice as the police investigation enters its third week."This horrific event should have never happened," the mother said.
"Hopefully one day justice will be served."
Police previously described the incident as deliberate and unprovoked.
"It is callous and cowardly," Detective Inspector Paul Dalton said.
"In 35 years of being a police officer, I've never come across a job where a child this young in these circumstances has been attacked."
The man was wearing glasses, a blue shirt, cargo shorts and a black hat.
He is believed to be aged about 30 to 40.
A GoFundMe page for the infant has so far raised more than $150,000.
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