  We consider the expansion of small-$x$ resummed DGLAP splitting functions at
next-to-leading logarithmic (NLL) accuracy to four-loop order, namely
next-to-next-to-next-to-leading order (N$^3$LO). From this, we extract the
exact LL and NLL small-$x$ contributions to the yet unknown N$^3$LO splitting
functions, both in the standard $\overline{MS}$ scheme and in the $Q_0
\overline{MS}$ scheme usually considered in small-$x$ literature. We show that
the impact of unknown subleading logarithmic contributions (NNLL and beyond) at
N$^3$LO is significant, thus motivating future work towards their computation.
Our results will be also needed in future to match NLL resummation to N$^3$LO
evolution. In turn, we propose an improved implementation of the small-$x$
resummation and therefore release a new version of the resummation code (HELL
3.0) which contains these changes.
