ROME—Italian Prime Minister Giuseppe Conte faces a showdown with his coalition partner and former premier Matteo Renzi this week that could bring down his government even as it struggles to contain the COVID-19 pandemic. Long-strained relations between the two men have deteriorated markedly in recent weeks, with Renzi calling for radical changes to plans to reboot the economy while also demanding that Conte cede his control over the secret services. The prime minister has resisted the pressure, saying last week that he was prepared to confront Renzi in parliament, effectively challenging him to act on his recent threats and abandon the coalition, thereby triggering an ill-timed crisis. Renzi told Il Messaggero newspaper over the weekend that he wouldn’t back down and dismissed suggestions that Conte could persuade a limited number of opposition politicians to prop up the government if his small Italia Viva party walked away. “If he has decided …