Public space, the conference's board of directors said, "is not the monopoly or privilege of any particular ideological group."
The expressed support was one of five areas addressed in a message sent July 13 to members of religious communities in Cuba from the directors.
Saying they are "in communion" with the Catholic bishops of Cuba regarding protests that erupted July 11 in cities across the country, the religious leaders explained that "we cannot close our eyes or look the other way, as if nothing were happening."
"We who are responsible for accompanying consecrated religious life in Cuba, welcome, with deep respect and interest, the cries and hopes manifested by the people who have gone out to protest on the street throughout the country," the directors said.
"As consecrated religious, we experience these events from a perspective of faith, and we recognize God's voice in the people's grievances. The people who came out into the streets are not delinquents; they are everyday Cuban people who found a way to express their discontent," the message said.
Thousands of people joined the demonstrations, protesting the unprecedented scarcity of essentials and the rising death toll caused by COVID-19. Clashes broke out between police and protesters, resulting in a few minor injuries, and some arrests were made.
The religious leaders called for the release of those who were "unjustly arrested for simply exercising their right to protest."
They also demanded that social media and cellphone service be restored. "This (blocking) increases the uncertainty and consternation of a people that already feels overwhelmed by dire economic, health and social situations," the leaders said.
The message urged all people to avoid "the trap of using violence as a way to impose one's own version of the truth."
"We are worried that due to a lack of capacity for dialogue and listening, the government will attack, repudiate, persecute and condemn people who think differently and express their views in public," the leaders said.
Finally, they encouraged all parties to "listen to each other to find solutions for the root causes behind these protests." Doing so, they said, will lead to healing.