  We derive exact relations between certain integrals of the conserved flavor
current four point function in 4d $\mathcal{N}=2$ conformal field theories
(CFTs) and derivatives of the mass deformed sphere free energy, which can be
computed exactly for gauge theories using supersymmetric localization. For
conformal gauge theories with flavor groups of rank greater than one, there are
at least two such integrated constraints, which can then be combined with the
numerical conformal bootstrap to bound CFT data as a function of the
complexified gauge coupling $\tau$. We apply this strategy to the case of
$SU(2)$ conformal SQCD with flavor group $SO(8)$, where we compute bounds on
unprotected scaling dimensions as a function of $\tau$ that match the free
theory limit, and exhibit the expected mixing between the action of the
$SL(2,\mathbb{Z})$ duality group and $SO(8)$ triality.
