Can dual gravity be reconciled with E11?

We extend a recently proposed formulation of dual gravity to the case of eleven-dimensional supergravity. The supersymmetric action corresponding to this alternative formulation is given, and it is shown that it leads to a set of first-order duality relations from which all second-order equations of motion follow as integrability conditions. On top of the fields corresponding to the conjectured E11 symmetry the action features Stueckelberg gauge fields that facilitate the realization of arbitrary symmetries on the dual graviton. However, there is no gauge-fixing that allows to eliminate the Stueckelberg fields. Therefore, E11 by itself is not a symmetry of eleven-dimensional supergravity but has to be extended at least by the Stueckelberg symmetries.