Meghan Markle and Prince Harry 'Taking Their Time' with New Archewell Foundation

"They are excited to see it all come together," a source tells PEOPLE. "It's what they have always dreamed of doing together"

Prince Harry and Meghan Markle. Photo: Chris Jackson/Getty

Prince Harry and Meghan Markle are moving forward with their new organization, Archewell, and the trademark filings are "live and pending," says a source.

Reports out of the U.K. that the trademarks had been rejected are "false," the source tells PEOPLE, adding that the filings "are going through the customary and normal process."

In fact, the Duke and Duchess of Sussex — who began working on the new foundation three months ago, when they relocated to Los Angeles with 1-year-old son Archie — have been quietly focused behind the scenes on further developing Archewell amid the rapidly changing social landscape.

"They have had the opportunity to speak with some incredible thought leaders, and those conversations have been deeply educational and helpful in identifying the work they want to do with Archewell," says the source. "They are applying all their learnings to the future of the organization."

Prince Harry and Meghan Markle.

When the couple announced the foundation in April, they explained the origin of the name, saying in a statement: “Before SussexRoyal came the idea of ‘arche’ — the Greek word meaning ‘source of action.' We connected to this concept for the charitable organization we hoped to build one day, and it became the inspiration for our son’s name. To do something of meaning, to do something that matters. Archewell is a name that combines an ancient word for strength and action, and another that evokes the deep resources we each must draw upon. We look forward to launching Archewell when the time is right.”

Meghan Markle and Archie. The Duke of Sussex/@SaveChildrenUK

Given the massive upheaval from both the COVID-19 pandemic and the racial justice movement sparked by the killing of George Floyd, Meghan and Harry have refocused their efforts.

"Nobody could have counted on the world turning upside down," says the source. "As many others have, they have been moved to get their voices behind supporting both the COVID-19 response and Black Lives Matter.

"But that’s not separate or a diversion from the values and mission that will be reflected in Archewell," the source continues. "You’ll see what they’re focused on now will also be reflected in the core of the organization. They are excited to see it all come together. It’s what they have always dreamed of doing together."

On June 4, Meghan addressed the Black Lives Matter movement in a surprise commencement speech for her alma mater, Immaculate Heart High School in L.A.

Meghan Markle addressing the Immaculate Heart Class of 2020. Meghan, the Duchess of Sussex

"I wanted to say the right thing, and I was really nervous that I wouldn’t, or that it would get picked apart, and I realized the only wrong thing to say is to say nothing," she shared in the video address. "Because George Floyd's life mattered and Breonna Taylor's life mattered and Philando Castile's life mattered and Tamir Rice's life mattered. And so did so many other people whose names we know and whose names we do not know. Stephon Clark, his life mattered."

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Addressing the class of 2020, she said, "You are going to have empathy for those who don't see the world through the same lens that you do, because with as diverse and vibrant and open-minded as I know the teachings at Immaculate Heart are, I know you know that Black lives matter.

"So I'm already excited for what you are going to do in the world. You are equipped, you are ready, we need you and you are prepared. I am so proud to call each of you a fellow alumni. And I'm so eager to see what you're going to do."