  A system of linear equations in $\mathbb{F}_p^n$ is \textit{Sidorenko} if any
subset of $\mathbb{F}_p^n$ contains at least as many solutions to the system as
a random set of the same density, asymptotically as $n\to \infty$. A system of
linear equations is \textit{common} if any 2-colouring of $\mathbb{F}_p^n$
yields at least as many monochromatic solutions to the system of equations as a
random 2-colouring, asymptotically as $n\to \infty$. Both classification
problems remain wide open despite recent attention.
  We show that a certain generic family of systems of two linear equations is
not Sidorenko. In fact, we show that systems in this family are not locally
Sidorenko, and that systems in this family which do not contain additive tuples
are not weakly locally Sidorenko. This endeavour answers a conjecture and
question of Kam\v{c}ev--Liebenau--Morrison. Insofar as methods, we observe that
the true complexity of a linear system is not maintained under Fourier
inversion; our main novelty is the use of higher-order methods in the frequency
space of systems which have complexity one. We also give a shorter proof of the
recent result of Kam\v{c}ev--Liebenau--Morrison and independently Versteegen
that any linear system containing a four term arithmetic progression is
uncommon.
