Reigning Open world champion Reg Bamford, of South Africa, appears to be among the ones to beat. He is one of 11 players to go through the first three days of play unbeaten in the eight blocks of 10.
Others who have won all six matches are three New Zealanders, six Egyptians and one Englishman.
The New Zealanders are national champion and holder of seven national titles Duncan Dixon, 25, of Bay of Plenty, Phillip Drew, 27, of Auckland, and Jenny Clarke, of Christchurch.
Another seven players, including two New Zealanders, Chris Clarke, of Christchurch, and John Christie, of Morrinsville, have won five of the best-of-three matches.
Women's world champion Rachel Rowe, of England, has four wins alongside the world under-21 champion from last week, Joshua Freeth, of Christchurch.
The field of 80 of the world's best is playing nine block matches over five days, with knockout competition to follow.
The top four from each of the 10 blocks will proceed to the knockout phase of the championship.
While one win will do the trick for Stephens, he emphasised the need to try to claim a further victory to ensure he would not have to face a top gun straight away.
"The higher you finish up the block, the weaker qualifiers you're likely to face from another block."
Stephens said three to four of them in Block B were "pretty even" going into today's matches.
"It could all drastically change tomorrow."
He had a dream start on day one on Saturday with two wins at the majestic Rotorua Croquet Club but Stephens had his reservations about the lawns.
"I didn't play that well and won. Some days you are scratchy and wonder how the hell you did it," he explained, saying the Rotorua lawns were "very bumpy" with balls sometimes leapfrogging other balls when struck.
"It's a hot spot of thermal activity so the shots bubble up quite a bit.
"You put your hand there and it feels warm. They've dug out domes but can't do much else about it."
Stephens said it didn't detract from the fact that the club, established in 1906, offered a stunning backdrop of government building gardens.
The clubhouse was last renovated in 1999 when the equipment shed was also built - with the pagoda-shaped roof - identical to the roof in the original building which had been altered in the intervening years.
The buildings are registered with the Rotorua Historical Trust, considering their location in front of the Rotorua Museum and the thermal belt.
Mt Maunganui, with its six lawns, was one of the best in the country while Katikati and Whakatane croquet clubs were "quite acceptable".
He plays at Whakatane today and has had "no complaints" about it.
His daughter, Erica Stephens, in her first golf croquet Open worlds, yesterday lost to John van der Touw, of Australia, 7-3, 7-6, and Marwan El Sabarouti, of Egypt, 7-3, 7-2, at Rotorua club to remain winless.
"It's tough for women. She did take a game off an Egyptian [Hisham Abu Osbaa] on day one," he said of Erica, who lost 7-4, 3-7, 4-7.
"She lacks consistency."
