  We prove an analogue statement to an estimate by De Lellis-M\"uller in
$\mathbb{R}^3$ on the standard Minkowski lightcone. More precisely, we show
that under some additional assumptions, any spacelike cross section of the
standard lightcone is $W^{2,2}$-close to a round surface provided the
trace-free part of a scalar second fundamental form $A$ is sufficiently small
in $L^2$. To determine the correct intrinsically round cross section of
reference, we define an associated $4$-vector, which transforms equivariantly
under Lorentz transformations in the restricted Lorentz group. A key step in
the proof consists of a geometric, scaling invariant estimate, and we give two
different proofs. One utilizes a recent characterization of singularity models
of null mean curvature flow along the standard lightcone by the author, while
the other is heavily inspired by an almost-Schur lemma by De Lellis-Topping.
