But some questioned whether the fungus was, indeed, the pukurau.
An entry on fungi in the government's online Te Ara: The Encyclopedia of New Zealand, refers to the pukurau as a puffball mushroom.
While it confirmed the white basket fungus was used by Maori as a food source, it was referred to by an entirely different traditional name.
"Maori traditionally used several … fungi as food, including the … basket fungus or kopurawhetu (Ileodictyon cibarium). Puffballs such as pukurau (Calvatia gigantea) were also used to staunch bleeding, and as anaesthetics," the entry states.
Fleming, who chaired the Waipukurau Rotary Club's book committee which commissioned Parsons to write the book in 2000, said he had spoken to the author to clear up any confusion.
"Patrick has heard that the giant puffball has been referred to as the pukurau but he does not believe that to be true. His research and talking with local Maori substantiates what he has written in his book."
That was backed up by Te Taiwhenua o Tamatea chairman, Dr Roger Maaka, who said as far as he was concerned the lattice-like ileodictyon fungus was the pukurau.
Various authorities on the subject said they were aware of both fungi being referred to as pukurau by Maori.
Peter Buchanan, science team leader at Manaaki Whenua Landcare Research, said pukurau was most typically used to describe puffballs in the genera Calvatia and Lycoperdon, which were considered edible when juvenile and were white internally.
However, Ileodictyon also had a great variety of Maori names, including kopurawhetu, tutaewhatitiri, and whareatua, "but occasionally also pukurau", he said.
Acclaimed Kiwi wildlife photographer Rod Morris has pictures of puffballs on his website which he refers to as pukurau. But he was aware of Ileodictyon going by at least 10 Maori names, including pukurau.
Dr Ocean Mercier a senior lecturer at Te Kawa a Maui, Victoria University's School of Maori Studies, said there were precedents of Maori using the same name to classify different things - such as Parearau being used to describe both Venus and Jupiter, despite their also having individual names.
"Perhaps puffballs and white basket fungus were grouped together for a particular reason, and both referred to as pukurau."
CHB mayor Alex Walker said she had no reason to doubt the Ileodictyon specimen she had on her desk was anything but an example of the pukurau.
"The specimen I have was found by council staff some time ago, and then dried with a ball of paper inside it so it kept its shape.
"I had thought it was the pukurau, after which the town was named.
"I haven't had anyone tell me otherwise, but I also haven't questioned it. - I took it on face value as I had never seen them before."
Noreen Farrell, from Waipawa, who contacted us to say she had five or six growing on the wooden fence in her backyard, said she too had never seen them in her 20 years living in CHB.
"I had no idea what it was. It looked like white netting. I'd never seen it before."
New CHB resident Lisa Wilson-Henare said she didn't know what to make of them when she found them in her garden on Hatuma Rd in Waipukurau.
"At first I thought it was an animal's insides. Then I thought it looked like coral."
Rosemairi Knowles, from Palmerston North, even heard about the story. She had seen them before, but never in her garden until last month.
"I have never seen them before in my 72 years.
"They start from what looks like a mushroom starting to grow, and obviously outgrow their 'incubator' and it splits open and out pops the basket like a baby bird, leaving behind the perfectly-formed lattice outline of the basket and some fluid.
"They are spongy to feel [and] last for days. Fascinating," she said.