Tshisekedi, 55, hopes the elections will hand him the presidential prize that eluded his late father Etienne, who founded the DRC's mainstream opposition party, the Union for Democracy and Social Progress (UDPS), in 1982. "These results have nothing to do with the truth of the ballot box".
A statement by more than 300 civil society organizations said people should be ready to protest. As Congo anxiously awaits the outcome of the presidential election, many in the capital say they are convinced that the opposition won and that the delay in announcing results is allowing manipulation in favor of the ruling party.
The other main opposition candidate, former oil executive Martin Fayulu - who was tipped as favourite in the few pre-election opinion polls - also sounded a conciliatory tone on Tuesday. The church replied by saying that only the release of false results would incite an uprising.
He is barred from serving three consecutive terms, but during more than two years of election delays many Congolese feared he'd find a way to stay in office.
The head of the Independent National Electoral Commission (Commission Électorale Nationale Indépendante: CENI), Corneille Nangaa, announced on 6 January that provisional results from the DRC's 30 December presidential, legislative, and provincial elections would not be announced that day (as scheduled). He is known as the "people's soldier" for leading protests against President Kabila. Earlier, he said that the results were "not negotiable" and that election officials should not "disguise the truth".
The Kabila family has led Congo for more than 20 years.
The UDPS described Tshisekedi as the presumptive victor with Secretary General Jean-Marc Kabund saying, "the two personalities have an interest in meeting to prepare for a peaceful and civilized transfer of power".More news: Camera catches man licking doorbell
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Congo's government has rejected Western pressure over the long-delayed election, calling it interference. The discrepancy sets the stage for a possible standoff between the Congolese government and the Church, which deployed 40,000 election observers and is considered one of Congo's most influential and trusted institutions.
Election observers reported a number of irregularities during the vote and the opposition alleged it was marred by fraud.
The mineral-rich country has never had a peaceful transition of power since it gained independence from Belgium in 1960. The machines were the focus of much concern, with the opposition and observers saying they could open the door to manipulating results.
The party behind the other top opposition candidate, Tshisekedi, said it suspects the electoral commission to be working in concert with Congo's ruling party under "purely political motivations".
Opposition candidate Martin Fayulu speaks to the press at his headquarters in Kinshasa, Congo, Jan. 10, 2019. As the electoral commission met this week, anti-riot police moved into place outside.
Nangaa said that, instead, the results announcement would be delayed until "next week", claiming that around half of the votes remained uncounted.