Thanks, Lilyflo.
The present campaign began in 2013 under the name Hares are Running and was spurred by the Succession to the Crown Act 2013 which removed sex discrimination from the succession to the crown. Its previous lead campaigner was the Countess of Clancarty, whose husband's earldom is set to become extinct even though the couple has a daughter.
People related to the Royal Family who have signed their names to the petition for equal primogeniture are the then Lady Serena Linley (the ex-wife of the current Earl of Snowdon), Melanie Cable-Alexander (the mother of the 1st Earl of Snowdon's son Jasper Cable-Alexander), and the Marchioness of Milford Haven (the wife of a cousin of Prince Philip).
Charlotte Carew Pole, who renewed the campaign under the name of Daughters' Rights, was born as a middle-class commoner, so she is not in line to inherit any title herself with or without a change to equal primogeniture.
She was inspired to become an activist against male primogeniture because of the behavior of some of her husband's aristocratic relatives when she gave birth to a daughter:
Charlotte Carew Pole, founder of Daughters' Rights, a campaign against [male] primogeniture, describes her aristocratic family's reaction when her first child, a daughter, was born. Of course, they were happy to have the baby. But, she said, there were frequent comments about what a shame it was that the child couldn't inherit and how Charlotte and her husband "must try again." Can you imagine the heartbreak of seeing that reaction to the birth of your first child? Charlotte says she had not, until that moment, seen that people would actually feel "a girl might not be the right baby to have."
https://www.doublexeconomy.com/post...care-about-gender-exclusion-among-the-peerage
After seven miscarriages and two rounds of IVF Charlotte Carew Pole was “absolutely thrilled” when she gave birth to her daughter, Jemima. Yet the penny soon dropped that, for some, Jemima was not good enough.
“It was definitely a thing,” she said. “‘Congratulations, what a shame it wasn’t a boy’, or ‘How quickly can you have another?’” were some of the comments she received.
“There was a general expectation that I must keep pumping them out until a boy arrived. And all because I married a man who will inherit a title.”
https://www.thetimes.co.uk/article/ladies-first-in-tory-plan-to-abolish-primogeniture-3qznb7j5l