Decidability of equational theories of coverings of semigroup varieties

For every proper semigroup variety $\mathfrak X$, there exists a semigroup variety $\mathfrak Y$ satisfying the following three conditions: (1) $\mathfrak Y$ covers $\mathfrak X$, (2) $\mathfrak X$ is finitely based then so is $\mathfrak Y$, and (3) the equational theory of $\mathfrak X$ is decidable if and only if so is the equational theory of $\mathfrak Y$. If $\mathfrak X$ is an arbitrary semigroup variety defined by identities depending on finitely many variables and such that all periodic groups of $\mathfrak X$ are locally finite, then one of the following two conditions holds: (1) all nilsemigroups of $\mathfrak X$ are locally finite and (2) $\mathfrak X$ includes a subvariety $\mathfrak Y$ whose equational theory is undecidable and which has infinitely many covering varieties with undecidable equational theories.