Properties of finite unrefinable chains of ring topologies

Let $R(+,\cdot)$ be a nilpotent ring and $(\mathfrak M,<)$ be the lattice of all ring topologies on $R(+,\cdot)$ or the lattice of all such ring topologies on $R(+,\cdot)$ in each of which the ring $R$ possesses a basis of neighborhoods of zero consisting of subgroups. Let $\tau$ and $\tau'$ be ring topologies from $\mathfrak M$ such that $\tau=\tau_0\prec_\mathfrak M\tau_1\prec_\mathfrak M\dots\prec_\mathfrak M\tau_n=\tau'$. Then $k\leq n$ for every chain $\tau=\tau'_0<\tau'_1<\dots<\tau'_k=\tau'$ of topologies from $\mathfrak M$, and also $n=k$ if and only if $\tau'_i\prec_\mathfrak M\tau'_{i+1}$ for all $0\leq i<k$.