Conde is proposing a change to the constitution to codify gender equality and introduce other social reforms. But his opponents fear the real motive is to reset presidential term limits, allowing the 82-year-old president to run for a third spell in office later this year - a scenario his government has not discounted.
Shortly after voting began at 08:00 GMT, young people attacked police deployed outside a polling station in a school in Ratoma, a suburb of the capital, Conakry, according to an AFP news agency reporter and other witnesses. In another school nearby, voting equipment was vandalised.
Since October, Guineans have protested en masse against the possibility of Conde extending his grip on power. At least 31 people and one gendarme have been killed to date, according to reports.
Originally scheduled for March 1, Conde postponed the referendum late last month after international criticism of some 2.5 million dubious names on the country's electoral roll. Some 7.7 million people were on the register, out of a total population about 13 million people.
The government says it has now scrubbed the problematic names, after an expert team from the West Africa bloc ECOWAS urged doing so last week. But Guinea's embattled opposition still doubts the vote's credibility.
Cellou Diallo, a former prime minister and the head of the leading opposition party, the UFDG, said the process of cleaning up the electoral roll had been opaque.
"It is an electoral masquerade," he said, adding that the roll did not reflect the electorate. His party, as well as the other large opposition parties, are boycotting both the referendum and the parliamentary election.
In a deeply polarised political environment, opposition figures have also promised to stop the votes from taking place.
Sidya Toure, the head of the opposition UFR party, said Conde wants to "convince the international community he is open to criticism" while pursuing a third term anyway.
Conde in 2010 became Guinea's first democratically-elected president. He was returned to office by voters in 2015 for his second and final five-year term under the current constitution, but critics say he has become increasingly authoritarian, citing arrests of protest leaders.
The draft constitution would also limit presidential terms to two but extend the length of the term to six years, potentially enabling Conde to govern for another 12 years.
Guinea's government argues the new constitution would usher in badly-needed changes in the country, especially for women. These would include banning female genital mutilation and under-age marriage and giving spouses equal rights in a divorce.
Conde has not denied that he might use the new constitution to seek another term when his second term runs out this year. He told French media last month that there was "nothing more democratic" than holding a referendum and it was up to his party to determine whether he would run again.
Amid concerns about the fairness of Sunday's vote, there is also increasing concern about the spread of the new coronavirus. Guinea has confirmed two cases to date.
"I have the impression our country is taking things lightly," said Amadou Oury Bah, a banker and politician.
The governing party called for people to vote but also insisted on the strict respect for health measures.