Equality of V(A∞) and the pseudo-regularization of V(A)

Determine whether the variety V(A∞) generated by the bisemilattice A∞ (constructed from a bounded lattice A by adjoining an element ∞ with operations a · ∞ = 0_A and a + ∞ = 1_A) coincides with the pseudo-regularization \overline{V(A)} of the lattice variety V(A).

Background

From a bounded lattice A with bounds 0_A and 1_A, the authors construct a bisemilattice A∞ by adjoining a new element ∞ and defining a * ∞ = 0_A and a + ∞ = 1_A for all a ∈ A. They show that t(x,y) = x + xy is a pseudopartition operation on A∞ but not a partition operation.

It follows that V(A∞) is contained in the pseudo-regularization \overline{V(A)}; however, it is unclear whether equality holds, i.e., whether A∞ already generates the entire pseudo-regularization of V(A).