The Peano-circle conjecture for zippers

Let $M$ be a hyperbolic 3-manifold, let $Z^\pm$ be a zipper or P-zipper for $M$, and let $S^1_{\mathrm{univ}}$ be its universal circle. The group $\pi_1(M)$ acts on $S^1_{\mathrm{univ}}$. The Peano-circle conjecture. There is a continuous $\pi_1(M)$-equivariant surjection

$$S^1_{\mathrm{univ}}\longrightarrow S^2_\infty.$$

Such a map would generalize the known Peano curves associated with universal circles from several existing constructions, including surface bundles and certain flows, and is proposed for arbitrary zippers.