Unique-factorisation conjecture for germ-like generalized power series

Let $K((\mathbb{R}^{\leq 0}))$ be the ring of generalized power series, let $J$ be the germ ideal, and let $v_J$ denote the order-value. A series $a\in K((\mathbb{R}^{\leq 0}))$ is germ-like if either $\operatorname{ot}(a)=v_J(a)$, or $v_J(a)>1$ and $\operatorname{ot}(a)=v_J(a)+1$. Germ-like unique-factorisation conjecture. Every non-zero germ-like series in $K((\mathbb{R}^{\leq 0}))$ admits a unique factorisation into irreducibles. The supplied context establishes existence of irreducible factorisations for germ-like series and uniqueness when the order-value is at most $\omega^3$; the assertion without that bound remains open.