Prince Harry didn't hesitate, however.
The act will only further endear the couple to the New Zealand public, as the tour has been dotted with several casual vignettes, such as dressing down for a cafe meeting to talk about mental health.
After her high-five, Prime said spoke with the Prince about the lack of female representation in Pacific Parliaments while the Prince talked about the role men could play to address the imbalance.
Prime worried her name would be scrubbed from the Governor-General's list of future events, until she saw the royal high-five was included in the event's official video.
Prime further knew one of the charities the Duke and Duchess had nominated for donations in lieu of wedding presents worked to alleviate period poverty in India, so when she met Meghan Markle she talked about the work of Tukau Community Fund and MyCup NZ to distribute menstrual cups free of charge to Northland women.
''I raised it with the Duchess because she has spoken about period poverty before. Some people consider it embarrassing or taboo, but she has lifted the lid on it and made it a global topic,'' Prime said.
''I thanked her for her work and we had a little exchange about it. When it was time for her to go on she turned around, squeezed my wrist and said something along the lines of 'keep it up' or 'you're doing great work' ... I was too star-struck to remember exactly.''
Prime said Tukau's initiative had so far provided re-usable menstrual cups to 2400 women in Kawakawa, Moerewa and Kaikohe, freeing them from the expense of buying sanitary products.
''That's huge when you think about the financial and environmental impact, and the empowerment,'' she said.
The project had its genesis during last year's election campaign when Prime was shocked to learn how many girls were missing school every month because they couldn't afford sanitary products.
Her sister, Season-Mary Downs, launched a fundraiser and started giving away the cups, which last 10 years and normally cost $25. Women who could afford the cups gave a koha which then paid for a cup for someone who couldn't.