Devaney and Krych showed that, for the exponential family $\unicode[STIX]{x1D706}e^{z}$, where $0\,<\,\unicode[STIX]{x1D706}\,<\,1/e$, the Julia set consists of uncountably many pairwise disjoint simple curves tending to $\infty$. Viana proved that these curves are smooth. In this article, we consider quasiregular counterparts of the exponential map, the so-called Zorich maps, and generalize Viana’s result to these maps.