IS QHA EQUATIONALLY DEFINABLE?

A quasi-Hilbert algebra is an algebra ⟨A; →, 1⟩ of type ⟨2, 0⟩ satisfying the following collection of identities and quasi-identities:

x → x ≈ 1
x → y ≈ 1 & y → x ≈ 1 ⊃ x ≈ y
x → (y → x) ≈ 1

x_{1} → (x_{2} → x) ≈ 1 & x_{1} → (x_{2} → (x → y)) ≈ 1 ⊃ x_{1} → (x_{2} → y) ≈ 1.

Clearly the class QHA of all quasi-Hilbert algebras is a quasivariety.

Problem: Is QHA a variety? If QHA is equationally definable, provide an axiomatisation.

By results of Bou et al, there is some algebraic evidence to suggest that QHA is a variety.