The tall young man, the head of a major LGBT civil rights organization, looked out at his audience and asked what seemed, in 2011, like a preposterous question. “Is it possible,” he wanted to know, “that in the years to come we will be able to declare the movement over? That we will have reached, at long last, a time in which our goals will have been achieved?”

Many in the audience, which largely consisted of LGBT activists, scratched our heads and smirked. The movement over? It seemed impossible to imagine that a day might come when the work of the civil rights movement for LGBT people could ever be considered complete.

And yet, the last few years have left many Americans dizzy at the speed of progress, whether gay, straight, transgender or cisgender (an antonym of “trans” that applies to those comfortable with the gender they were assigned at birth). At times it’s been hard even to keep track of the string of marriage-equality victories. Marriage equality is now the law in 37 states; in six others, courts have overturned the ban on same-sex marriage, but those rulings are being appealed. And the Supreme Court is expected to resolve the legal battle over marriage equality by June.

Is it time to furl the rainbow flags and head home? The answer, alas, is no.

In an effort to measure the emerging cultural gap on LGBT issues, GLAAD — where I serve on the board of directors — commissioned a Harris poll last summer to measure the level of self-reported comfort with various aspects of LGBT life among straight, cisgender Americans. The results, to say the least, are sobering.

About 1 in 3 Americans reported feeling uncomfortable attending a same-sex wedding. A total of 43 percent said they would feel some discomfort bringing a child to a same-sex wedding. And 27 percent said they’d feel uncomfortable simply looking at a gay co-worker’s wedding picture. These numbers remain fairly close whether the state has marriage equality laws on its books or not.

The statistics are just as disheartening when it comes to perceptions of LGBT people as parents. About 1 in 3 Americans, according to the Harris poll, believes that a household with two married same-sex individuals, regardless of gender, is somewhat or very harmful to children. Americans are four times more uncomfortable with a gay woman supervising children than a straight one.

The numbers, unsurprisingly, are even more discouraging when it comes to perceptions of transgender individuals. About 40 percent of Americans show levels of discomfort with a transgender person supervising children — more than 10 percentage points higher than if the adult were a gay mom or dad.

The Harris poll shows even higher discomfort in the South with nearly every LGBT issue.

It’s been said many times that you can’t legislate morality, and that’s true. But it’s just as true that you can’t legislate openheartedness. We have reached a moment in our history when our laws, as far as LGBT citizens are concerned, may have become more tolerant than we are. And that should give everyone pause.

There is plenty more legal work ahead, in the courts and in the statehouses. But in some ways, the more difficult challenge has barely been addressed: reducing the level of discomfort among straight and cisgender people so that the men and women who increasingly enjoy legal protections in this country can also be accepted, respected and loved.

The battle to change the laws is nearing its end. The battle to open people’s hearts is just beginning.