Cardinality of the interval between full and nonempty power-semiring varieties

Determine the cardinality of the interval \([\mathsf{V}(\mathcal{P}^{+}(S)),\mathsf{V}(\mathcal{P}(S))]\) in the lattice of semiring varieties for an arbitrary semigroup \(S\).

Background

For any semigroup S, the nonempty power semiring $\mathcal{P}^{+}(S)$ is contained in the full power semiring $\mathcal{P}(S)$, inducing an inclusion between their generated varieties. The problem asks how many semiring varieties can occur between these two endpoints.

The paper answers the question for the multiplicative reduct of $S_7$: the interval between $\mathsf{V}(\mathcal{P}^{+}(S_7,\cdot))$ and $\mathsf{V}(\mathcal{P}(S_7,\cdot))$ has the cardinality of the continuum. The general problem for arbitrary semigroups remains unresolved.