Actually, thinking about it, I don't think I need loops to prove it.

This spider can't be made into a blob where all the arrows point in the same direction,

\$X^{op} \otimes Y \nrightarrow Z^{op} \otimes W. \$

Edit: Therefore the assumption that,

>"horizontal v vertical" or "left to right" aspects of string diagrams – at least for profunctors – turn out to be irrelevant.

is incorrect. Directionality cannot be ignored and made irrelevant under topological moves.