 Solution

Let $w$ be the number of ones. The displayed output state assigns the outcome $(0,0)$ probabilityFor a constant string, $w=0$ or $N$, so this probability is one. For a balanced string, $w=N/2$, so it is zero. Under the promise, $(0,0)$ certifies a constant string, and any other possible outcome certifies a balanced string. A nonzero ordered-pair outcome must have $i<j$ and ${\hat{x}}_i-{\hat{x}}_j\ne 0$, which also certifies $x_i\ne x_j$ without separately reading either bit. This is the fact used by opposite-pair elimination for exact quantum balance testing.