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'Kane continues to show good signs, but isn't quite ready to jump on a plane and join us,' Black Caps coach Gary Stead said in a news release.
The master batsman has remained in New Zealand to work on his recovery from the injury he sustained in a recent series against Sri Lanka.
The US, who were guaranteed entry as co-hosts, pulled off a Super Over win against Pakistan in Dallas after beating fellow tournament debutants Canada.
Hazlewood's inexorable line and length, just on off stump and ducking away, was too much for the New Zealand top order.
New Zealand consolidated their position at the top of the World Test Championship standings with 36 points and a winning percentage of 75.
New Zealand dashed India's hopes in the semifinals of the 2019 edition and Williamson said he was not concerned that his side were still considered underdogs.
New Zealand occupy the last semifinal slot available but only by virtue of having a better net run rate than Pakistan and Afghanistan, with all three teams tied on eight points after eight games.
Both Pakistan and New Zealand have eight from as many games with one match remaining.
Tom Blundell will travel to India as cover, but Williamson will remain with the team with the hope of being available for the back end of group play next month.
Williamson missed wins over defending champions England and the Netherlands to work on his fielding and running between the wickets.
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