  We probe the conformal block structure of a scalar four-point function in
$d\geq2$ conformal field theories by including higher-order derivative terms in
a bulk gravitational action. We consider a heavy-light four-point function as
the boundary correlator at large central charge. Such a four-point function can
be computed, on the gravity side, as a two-point function of the light operator
in a black hole geometry created by the heavy operator. We consider
analytically solving the corresponding scalar field equation in a near-boundary
expansion and find that the multi-stress tensor conformal blocks are
insensitive to the horizon boundary condition. The main result of this paper is
that the lowest-twist operator product expansion (OPE) coefficients of the
multi-stress tensor conformal blocks are universal: they are fixed by the
dimension of the light operators and the ratio between the dimension of the
heavy operator and the central charge $C_T$. Neither supersymmetry nor unitary
is assumed. Higher-twist coefficients, on the other hand, generally are not
protected. A recursion relation allows us to efficiently compute universal
lowest-twist coefficients. The universality result hints at the potential
existence of a higher-dimensional Virasoro-like symmetry near the lightcone.
While we largely focus on the planar black hole limit in this paper, we include
some preliminary analysis of the spherical black hole case in an appendix.
