How long do you keep a stalling position?

I locate S/R levels in the daily and look for breakouts in the 15min - but as you all know sometimes there are false breakouts and the price stays near the “broken” S/R, unable to break through.
Do you get rid of such positions once they’re a few pips above breakeven?

There’s no such thing as a false breakout. You have to remember who’s drawing these lines. If you start expecting price to behave a certain way when it nears lines you’ve drawn on a chart, perhaps you’re not quite comfortable with your system.

I stay in my positions until my system’s rules indicate an exit.