Equality of T2 and the pseudo-regularization of SQ

Determine whether the variety T2 of commutative idempotent groupoids defined by the identity x (y · yz) = (xy · y) z coincides with the pseudo-regularization \overline{SQ} of the variety SQ of Steiner quasigroups (commutative idempotent groupoids satisfying xy · y = x).

Background

The variety SQ consists of Steiner quasigroups, i.e., commutative idempotent groupoids satisfying xy * y = x. Its regularization SQ is strictly contained in the pseudo-regularization \overline{SQ}, which is defined by identities (P1)–(P4) with the pseudopartition operation t(x,y) = xy * y.

Prior work shows that the variety T2, consisting of commutative idempotent groupoids satisfying x (y * yz) = (xy * y) z, is contained in \overline{SQ} and is different from SQ. It is unknown whether \overline{SQ} equals T2.