  We show that the average null energy condition implies novel lower bounds on
the scaling dimensions of highly-chiral primary operators in four-dimensional
conformal field theories. Denoting the spin of an operator by a pair of
integers $(k,\bar{k})$ specifying the transformations under chiral
$\frak{su}(2)$ rotations, we explicitly demonstrate these new bounds for
operators transforming in $(k,0)$ and $(k,1)$ representations for sufficiently
large $k$. Based on these calculations, along with intuition from free field
theory, we conjecture that in any unitary conformal field theory, primary local
operators of spin $(k,\bar{k})$ and scaling dimension $\Delta$ satisfy $\Delta
\geq \text{max}\{k,\bar{k}\}.$ If $|k-\bar{k}| > 4$, this is stronger than the
unitarity bound.
