  The four-point function of length-two half-BPS operators in $\mathcal{N}=4$
SYM receives non-planar corrections starting at four loops. Previous work
relied on the analysis of symmetries and logarithmic divergences to fix the
integrand up to four constants. In this work, we compute those undetermined
coefficients and fix the integrand completely by using the reformulation of
$\mathcal{N}=4$ SYM in twistor space. The final integrand can be written as a
combination of finite conformal integrals and we have used the method of
asymptotic expansions to extract non-planar anomalous dimensions and structure
constants for twist-two operators up to spin eight. Some of the results were
already know in the literature and we have found agreement with them.
