What is the ruling on depositing money in a branch of a bank that declares itself to be Islamic and invests funds according to Sharia (and Allah has permitted trade and forbidden usury), but whose actual commitment cannot be verified?

There is no objection to placing money in the Islamic transaction branches of interest-based banks, provided that they adhere to the provisions of Islamic law, and their investment channels are separate from the interest-based bank, and they are not used as a cover to attract people's money and exploit their emotions and aversion to usury. However, if these branches do not adhere to the provisions of Islamic law, or are used as a cover to attract the money of Muslims, then it is not permissible to deal with them. If you are in doubt about the dealings of such a branch, you should not place your money in it. You can instead place it in one of the trusted Islamic banks, or invest it in Mudarabah (profit-sharing) and similar forms of permissible investment.