Odd-dimensional complexes have zero zigzag signatures

Let ${\cal K}$ be a complex of odd dimension. A zigzag in ${\cal K}$ has a signature, recording the numbers of intersections of the zigzag with the relevant types of faces. Zero-signature conjecture. The signature of every zigzag in ${\cal K}$ is $(0,0)$. The claim concerns the uniform intersection behaviour of zigzags in odd-dimensional complexes; the supplied text gives no evidence that it has been proved or disproved.