  We show that for $p>1$ there is no $p$-cyclically monotone stationary
matching of two independent Poisson processes in dimension $d=2$. The proof
combines the $p$-harmonic approximation result from \cite[Theorem 1.1]{koch23}
with local asymptotics for the two-dimensional matching problem. Moreover, we
prove a.s. local upper bounds of the correct order in the case $p>1$, which, to
the best of our knowledge, are not readily available in the current literature.
