Conjecture on semilattice-like terms in bounded width algebras

Let $\mathbb{A}$ be an algebra of bounded width, and let $w,s$ be terms of its language. The conjectured identities are

$$w(x,x,y) \approx w(x,y,x) \approx w(y,x,x) \approx s(x,y)$$

and

$$s(x,s(x,y)) \approx s(s(x,y),x) \approx s(x,y).$$

Semilattice-like terms conjecture. Every bounded width algebra has terms $w,s$ satisfying these identities.

The conjecture is known in every spiral and in every majority algebra; if true, it would imply Bulatov's yellow connectivity property and related results by iteration. Its general case remains open in the supplied text.