  We consider the mixed Dirichlet-conormal problem for the heat equation on
cylindrical domains with a bounded and Lipschitz base $\Omega\subset
\mathbb{R}^d$ and a time-dependent separation $\Lambda$. Under certain mild
regularity assumptions on $\Lambda$, we show that for any $q>1$ sufficiently
close to 1, the mixed problem in $L_q$ is solvable. In other words, for any
given Dirichlet data in the parabolic Riesz potential space $\mathcal{L}_q^1$
and the Neumann data in $L_q$, there is a unique solution and the
non-tangential maximal function of its gradient is in $L_q$ on the lateral
boundary of the domain. When $q=1$, a similar result is shown when the data is
in the Hardy space. Under the additional condition that the boundary of the
domain $\Omega$ is Reifenberg-flat and the separation is locally sufficiently
close to a Lipschitz function of $m$ variables, where $m=0,\ldots,d-2$, with
respect to the Hausdorff distance, we also prove the unique solvability result
for any $q\in(1,(m+2)/(m+1))$. In particular, when $m=0$, i.e., $\Lambda$ is
Reifenberg-flat of co-dimension $2$, we derive the $L_q$ solvability in the
optimal range $q\in (1,2)$. For the Laplace equation, such results were
established in [6, 5, 7] and [14].
