Is the story of Abu Bakr al-Siddiq (may Allah be pleased with him) reciting during rukūʿ, and the revelation descending with the words: "Allah hears him who praises Him," and Abu Bakr responding with: "Our Lord, to You be all praise," authentic and established?

The story of "Rabbana wa lakal-hamd" (Our Lord, to You be all praise) has no basis. The fundamental principle regarding the actions and sayings in prayer is what the Prophet, peace and blessings be upon him, taught us, as in his saying: «Pray as you have seen me praying.» The prophetic traditions have clarified that when the Imam says: «Sami'a Allahu liman hamidah» (Allah hears those who praise Him), the congregants should say: «Rabbana wa lakal-hamd» or «Allahumma Rabbana lakal-hamd» (O Allah, our Lord, to You be all praise). The virtue mentioned for this is the forgiveness of sins if the saying of the people of the earth coincides with the saying of the people of heaven. This story was mentioned on "Al-Durar Al-Saniyyah" website among the widely spread unauthentic hadiths and described as "false."