Bounds for Banach-Mazur distances between some $C(K)$-spaces

We present several results providing lower bounds for the Banach-Mazur distance $$d_{BM}(C(K), C(L))$$ between Banach spaces of continuous functions on compact spaces. The main focus is on the case where $C(L)$ represents the classical Banach space $c$ of convergent sequences. In particular, we obtain generalizations and refinements of recent results from \cite{GP24} and \cite{MP25}. Currently, it seems that one of the most interesting questions is when $K = [0, \omega]$ is a convergent sequence with a limit and $L = [0,\omega]\times 3$ consists of three convergent sequences. In this case, we obtain $$3.53125 \leq d_{BM}(C([0,\omega]\times 3),C[0,\omega]) \leq 3.87513$$