hold out little, etc. hope
hold out hope
To maintain hope that something will happen or be the case, especially when it does not seem likely. Often used in negative constructions and with modifiers like "much," "any," "little," etc., before the word "hope." We're still holding out hope that our missing son will be found. I know that this legislation is right for the country, but I hold out little hope that it will make it through both houses of congress. They didn't hold out much hope for being approved for a mortgage, given their credit history.