Off-shell supersymmetric continuous spin gauge theory

We construct, for the first time, an off-shell supersymmetric continuous spin gauge theory in 4-dimensional Minkowski spacetime, both constrained and unconstrained Lagrangian formulations. As extension to on-shell description [1], we employ auxiliary field close algebra of supersymmetry transformations off-shell. The 4d $\mathcal{N}=1$ massless supermultiplet is then denoted by $(\mathrm{\Phi}, \mathrm{H} \,; \mathrm{\Psi})$, comprised a dynamical non-dynamical complex scalar fields $\mathrm{\Phi}$ $\mathrm{H}$, as well Dirac $\mathrm{\Psi}$. In particular, demonstrate that supermultiplet, limit, reproduces known $(\,{\scriptstyle 0}\,,\, {\scriptstyle 1/2}\,)$, all integer-spin supermultiplets $(\,s\,,\, s\, +\, 1/2}\,),\, s\geqslant1$, half-integer $(\, -\, 1/2}\,,\, s \,\,),\, s\geqslant1$.