Properties of finite unrefinable chains of ring topologies for nilpotent rings

Let $R$ be a nilpotent ring and let $(\mathfrak M,<)$ be the lattice of all ring topologies or the lattice of all ring topologies in each of which the ring $R$ possesses a basis of neighborhoods of zero consisting of subgroups. If $\tau_0\prec_\mathfrak M\tau_1\prec_\mathfrak M\dots\prec_\mathfrak M\tau_n$ is an unrefinable chain of ring topologies from $\mathfrak M$ and $\tau\in\mathfrak M$, then $k\leq n$ for any chain $\sup\{\tau,\tau'_0\}=\tau'_1<\tau'_2<\dots<\tau'_k=\sup\{\tau,\tau_n\}$ of topologies from $\mathfrak M$.