Just what is in store for Roman Catholic women under Pope Francis' leadership?

For now, that remains a mystery of the faith.

Will Francis Alter Roles, Rules Affecting Catholic Women?

The pontiff arrived Tuesday for a six-day visit to the United States, trying to shore up support for an American church whose numbers are trailing population growth. Polls show most of the country’s 51 million Catholic adults embrace Francis but disagree with some fundamental church teachings affecting women, who make up just over half of the denomination. Key areas of contention involve women's reproductive health and females' limited roles in the church’s male-dominated hierarchy.

Will Francis Alter Roles, Rules Affecting Catholic Women?

Shortly after becoming pope in March 2013, Francis called for "still broader opportunities for a more incisive female presence" – but made clear women priests would not be accepted anytime soon. Nor has he pushed for women's broader inclusion in positions of authority throughout the church, some Vatican observers say.

Will Francis Alter Roles, Rules Affecting Catholic Women?

"The optimism that I felt two years ago ... has not really been borne out by anything that has happened," said Kathleen Sprows Cummings, director of the University of Notre Dame’s . While she noted the pope appointed a woman to lead a Vatican missionary group and chose five others for an international theological commission, "it’s not the sweeping changes that signal women have a place at the table."

"The bottom line is if women ... are turned off by the church’s patriarchal culture, we’re going to lose future generations," agreed Thomas J. Reese, a Jesuit and member of the .

Cummings, a historian, teaches a course on feminism in the church. The young women in her classes don’t necessarily consider themselves feminists, she said, but, having grown up in contemporary America, they "assume all doors are open to them."

Her students wrote letters of protest several years ago when they learned of threats to revoke the right for girls to serve on the altar. The Vatican has permitted female servers since at least 1994 at the discretion of each diocese and their individual priests.

Just outside of Washington, in Arlington, Virginia, 15-year-old Annemarie Dougherty in January became one of the first female servers at St. Agnes Catholic Church. (The Arlington diocese was one of the last to approve female servers, in 2007.)

"A lot of it was that girls weren’t allowed to do what boys were allowed to do, and that frustrated me," the high school sophomore said. Serving on the altar has "made me more aware of what is going on in different parts of the Mass."

Her mother had encouraged Annemarie’s participation. "When I was young, I always thought it was a little unfair. I felt that women were so unimportant in the church," Sheila Dougherty explained. "You had those priests up there, the men doing the readings, the men collecting the money."

The latter two roles, and others, have long since opened up to women. Now both mother and daughter are enthusiastic about the Catholic Church and its spiritual leader.

"The new pope – he’s so cool," Annemarie Dougherty said.

He’s also a 78-year-old celibate man, which may impede his understanding of women – and their expectations and potential within the faith community, Reese speculated.

"Celibacy is a problem for priests not because they don’t have sex. It’s because they don’t have a wife" or daughters to question their assumptions and give advice, said Reese, who, as senior analyst for the , has written about . He noted that the former Jorge Bergoglio comes from Argentina, and "the Latin American male ideal is macho, paternalistic. You can’t ignore that."

But on the flip side, Reese pointed out, Francis has a long history of working well for and with women – first as a chemist, later as "a leader in the movement against human trafficking" in a country that has had a female president since 2007.

As pope, he has said "the church needs a new theology about women," Reese added. Francis has "invited the church into a conversation. That’s the important thing."

On his U.S. trip, the pontiff might expect discussion on several points of particular interest to women.

: Three-fourths (76 percent) of U.S. Catholics differ with, or disregard, the church’s ban on the use of artificial contraception, .

And a slim majority of U.S. Catholics (51 percent) say abortion should be legal in all or most cases, the in a recent poll. That percentage falls to 35 percent among regular churchgoers.

Francis recently announced a special Holy Year of Mercy, in which any priest can grant women forgiveness for abortion, considered a grave sin by the church. The observation begins December 8, a holy day honoring belief in the Virgin Mary’s immaculate conception of Jesus.