hold out 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.
See also: hold, hope, out
Farlex Dictionary of Idioms.

hold out little, etc. ˈhope (of something/that...)

,

not hold out any, much, etc. ˈhope

offer little, etc. reason for believing that something will happen: The doctors did not hold out much hope for her recovery.
See also: hold, hope, out
Farlex Partner Idioms Dictionary
See also:
  • hold out little, etc. hope
  • give weight to
  • give weight to (something)
  • give (something) weight
  • give some thought to
  • give thought to (something)
  • give (something) thought
  • change (one's) stripes
  • change one's stripes/spots, cannot