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Thanks a lot for linking this question! I am aware of the Mazur--Rubin reference, and would be specifically interested in what (if anything) is known unconditionally.
I would be by far the most interested in a better bound for every sublattice $L$ of $\mathbb{Z}^t$ (i.e. a lower bound for $\text{inf}_{L \subseteq \mathbb{Z}^t} \frac{2^t \det(L)}{\lambda_1 \cdot \ldots \cdot \lambda_t \cdot \text{Vol}(K)}$), but I would also be happy to hear about other things known about this problem.
Look at the decomposition group of the rational prime $p$ in $\text{Gal}(L_k/\mathbb{Q})$ not in $\text{Gal}(L_k/L_{k - 1})$. If the fixed field of the decomposition group is $E$, then there is no more splitting in the extension $L_k/E$. But we know that there is splitting in $L_k/L_{k - 1}$ by assumption, hence $E = L_k$ and the decomposition group is trivial, so $p$ splits completely.
I could be totally in the wrong here but I was hoping that (at least in my case where $[L : K]$ is bounded and $K$ is fixed) that the ineffective Brauer-Siegel theorem gives me a good lower bound for the residue, which can then be translated in a good bound on the Siegel zero.
