It appears this can happen already with the
macro average option. The statement needs some clarification, but I assume the precision and recall that are supposed to not bound the averaged F1 are themselves the same type of average.
Here's a simple example: $TP=TN=4$, $FP=1$, $FN=16$. Then
and so $F_1(1)=F_1(0)=0.32$, so the macro-average $F_1$ is also $0.32$. But the macro-averaged precision and recall are both $0.5$.