  We study the conformal bootstrap constraints for 4D $\mathcal{N}=1$
superconformal field theories containing a chiral operator $\phi$ and the
chiral ring relation $\phi^2=0$. Hints for a minimal interacting SCFT in this
class have appeared in previous numerical bootstrap studies. We perform a
detailed study of the properties of this conjectured theory, establishing that
the corresponding solution to the bootstrap constraints contains a
$\text{U}(1)_R$ current multiplet and estimating the central charge and
low-lying operator spectrum of this theory.
