Explicit Bits-back Coding (a.k.a. Free Energy Coding) applied to Gaussian mixtures
Apparently, no. This is a generalization where the relationship between $k$ and $i$ is stochastic (i.e. defined by a distribution and not deterministic).
To recover the original bits-back argument, we would need to pick a value $i_k$ for each possible value of $k$ and force $p(i=i_k \vert x, k)=1$ and 0 if $i \neq i_k$. Naturally, we would need to extend the support of $k$ to the same of $i$. In this particular example, $k \in \{0, 1, 2, 3\}$