  Integrated correlator of four superconformal stress-tensor primaries in
$SU(N)$ $\mathcal{N}=4$ super Yang-Mills (SYM) theory in the perturbative limit
takes a remarkably simple form, where the $L$-loop coefficient is given by a
rational multiple of $\zeta(2L+1)$. In this letter, we extend the previous
analysis of expressing the perturbative integrated correlator as a linear
combination of periods of $f$-graphs, graphical representations for loop
integrands, to the non-planar sector at four loops. At this loop order,
multiple zeta values make their first appearance when evaluating periods of
non-planar $f$-graphs, but cancel non-trivially in the weighted sum. The
remaining single zeta value, along with the rational number prefactor, makes a
perfect agreement with the prediction from supersymmmetric localisation.
