Subvariety intervals and limit varieties of zero extensions of mirror-word semirings

For the mirror word ${\mathbf{m}}_n=a_1\cdots a_n{a}_n\cdots a_1$, let ${\mathcal{V}}_n=\mathrm{V}(S({\mathbf{m}}_n))$ and ${\mathcal{W}}_n=\mathrm{V}(S({\mathbf{m}}_n{)}^0)$. We show that $[{\mathcal{W}}_n,{\mathcal{W}}_{n+1}]$ ($n\ge 1$) and $[{\mathcal{V}}_n,{\mathcal{W}}_n]$ ($n\ge 2$) have cardinality $2^{{\mathrm{\aleph }}_0}$. Every variety in $[{\mathcal{W}}_2,{\mathcal{W}}_{\mathrm{\infty }}]$ has infinite axiomatic rank. The unique limit subvariety of ${\mathcal{W}}_1$ is $\mathrm{V}(S{R}_6)$. For $n\ge 2$, we construct a distinct directed-path limit variety contained in ${\mathcal{W}}_n$ but not in ${\mathcal{V}}_n$.