Finitely based sets of 2-limited block-2-simple words

Let $\mathfrak A$ be an alphabet and $W$ be a set of words in the free monoid ${\mathfrak A}^*$. Let $S(W)$ denote the Rees quotient over the ideal of ${\mathfrak A}^*$ consisting of all words that are not subwords of words in $W$. A set of words $W$ is called {\em finitely based} if the monoid $S(W)$ is finitely based. A word $\bf u$ is called 2-limited if each variable occurs in $\bf u$ at most twice. A {\em block} of a word $\bf u$ is a maximal subword of $\bf u$ that does not contain any linear variables. We say that a word $\bf u$ is {\em block-2-simple} if each block of $\bf u$ involves at most two distinct variables. We provide an algorithm that recognizes finitely based sets of words among sets of 2-limited block-2-simple words. We also present new sufficient conditions under which a set of words is non-finitely based.