${\cal N}=1$ conformal duals of gauged $E_n$ MN models

We suggest three new ${\cal N}=1$ conformal dual pairs. First, we argue that the ${\cal N}=2$ $E_6$ Minahan-Nemeschansky (MN) theory with a $USp(4)$ subgroup of the $E_6$ global symmetry conformally gauged with an ${\cal N}=1$ vector multiplet and certain additional chiral multiplet matter resides at some cusp of the conformal manifold of an $SU(2)^5$ quiver gauge theory. Second, we argue that the ${\cal N}=2$ $E_7$ MN theory with an $SU(2)$ subgroup of the $E_7$ global symmetry conformally gauged with an ${\cal N}=1$ vector multiplet and certain additional chiral multiplet matter resides at some cusp of the conformal manifold of a conformal ${\cal N}=1$ $USp(4)$ gauge theory. Finally, we claim that the ${\cal N}=2$ $E_8$ MN theory with a $USp(4)$ subgroup of the $E_8$ global symmetry conformally gauged with an ${\cal N}=1$ vector multiplet and certain additional chiral multiplet matter resides at some cusp of the conformal manifold of an ${\cal N}=1$ $Spin(7)$ conformal gauge theory. We argue for the dualities using a variety of non-perturbative techniques including anomaly and index computations. The dualities can be viewed as ${\cal N}=1$ analogues of ${\cal N}=2$ Argyres-Seiberg/Argyres-Wittig duals of the $E_n$ MN models. We also briefly comment on an ${\cal N}=1$ version of the Schur limit of the superconformal index.