Bounded relational-width term conjecture for finite idempotent algebras

Let $\mathbb{A}$ be a finite idempotent algebra, meaning that every basic operation satisfies $f(x,\ldots,x)=x$. Bounded relational-width term conjecture. The algebra $\mathbb{A}$ has bounded relational width if and only if it has a ternary term $m$ and an associated binary term $s(x,y)\coloneqq m(x,x,y)$ satisfying

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

and

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

The source calls this a strong refinement and gives no resolution evidence.