Reducibility and irreducibility of monomial matrices over commutative rings

Let $R$ be a local ring with nonzero Jacobson radical. We study monomial matrices over $R$ of the form
$$ \left( \begin{smallmatrix} 0&\ldots&0&t^{s_n}\\ t^{s_1}&\ldots&0&0\\ \vdots&\ddots&\vdots&\vdots\\ 0&\ldots&t^{s_{n-1}}&0\\ \end{smallmatrix} \right), $$
and give a criterion for such matrices to be reducible when $n\leq 6$, $s_1\ldots,s_n\in\{0,1\}$ and the radical is a principal ideal with generator $t$. We also show that the criterion does not hold for $n=7$.