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Ardern attended a dinner with Google executives a few hours ago in Davos, Switzerland, where she told media she planned to try and raise the issue with Google executives.
Millane, who was holidaying in New Zealand, was killed in early December last year. She is believed to have died on the weekend of her 22nd birthday, just a day after she arrived in Auckland as part of a one-year solo OE. Her body was found in the Waitakere Ranges.
The identity of the man accused of the murder was suppressed but, in an email to people signed up to Google's "what's trending in New Zealand" newsletter, the search engine named the accused by saying they were trending.
Asked about raising the issue with Google executives at the dinner, Ardern said she would try - but said it was something that had been raised with Google already.
"It will really be a matter of us raising it again."
Justice Minister Andrew Little met with Google executives late last year to discuss the issue.
Google NZ's senior manager of public policy, Ross Young, said at the time the search engine received a notification about the name suppression four days after interim suppression was granted.
Asked why it had taken so long, he said: "I don't know."
"We respect New Zealand law and we will respond to court orders when we get them. As you'll appreciate, there are trillions of web pages, dynamic and active, on the internet at the moment."
Little said he would contact Google in the New Year for an update.
Ardern was careful not to overplay how much pressure she would be applying on Google at the function.
"It is a dinner, but I will try and take the opportunity to raise that with the most appropriate person, at a time that I can," Ardern told media this morning.
She stressed that it was just a dinner, with "something like 20 people".
"Often the conversations are quick snippets, depending on where you're sitting. But yes, I do intend to [raise the issue] anyway."