Trace reconstruction with varying deletion probabilities

In the trace reconstruction problem an unknown string ${\bf x}=(x0,\dots,x{n-1})\in{0,1,...,m-1}^n$ is observed through the deletion channel, which deletes each $xk$ with a certain probability, yielding a contracted string $\widetilde{\bf X}$. Earlier works have proved that if each $xk$ is deleted with the same probability $q\in[0,1)$, then $\exp(O(n^{1/3}))$ independent copies of the contracted string $\widetilde{\bf X}$ suffice to reconstruct $\bf x$ with high probability. We extend this upper bound to the setting where the deletion probabilities vary, assuming certain regularity conditions. First we consider the case where $xk$ is deleted with some known probability $qk$. Then we consider the case where each letter $\zeta\in {0,1,...,m-1}$ is associated with some possibly unknown deletion probability $q_\zeta$.