The strong $\pi$-regularity conjecture for pseudo $\pi$-polar $\pi$-UU rings

Let $R$ be a ring. A ring is pseudo $\pi$-polar if it satisfies the pseudo $\pi$-polar condition, and $\pi$-UU if every unit $u\in U(R)$ has $1-u^n\in J(R)$ for some integer $n\geq 1$. Strong $\pi$-regularity conjecture. Every pseudo $\pi$-polar $\pi$-UU ring is strongly $\pi$-regular, and hence periodic. This would extend the established equivalence between strongly $\pi$-regular pseudo $\pi$-polar rings and periodic rings, while the preceding example shows that pseudo $\pi$-polarity alone does not imply the $\pi$-UU property.