Lubin's conjecture on stable power series and Lubin–Tate formal groups

Let $K$ be a finite extension of ${\mathbb Q}_p$ with ring of integers ${\mathcal O}_K$, unit group $U_K$, and let ${\mathcal D}_0({\mathcal O}_K)$ denote the power series over ${\mathcal O}_K$ under consideration. For a power series $g$, let $\operatorname{Stab}_{{\mathcal O}_K}(g)$ be its stabilizer, and let $\partial_0$ denote the derivative at zero. Suppose that $g\in {\mathcal D}_0({\mathcal O}_K)$ is stable and noninvertible, with height equal to $[K:{\mathbb Q}_p]$, and satisfies

$$\partial_0\left(\operatorname{Stab}_{{\mathcal O}_K}(g)\right)=U_K.$$

Lubin's conjecture. Then $g$ is an endomorphism of some Lubin–Tate formal group $F(x,y)$ over ${\mathcal O}_K$, and

$$\operatorname{Stab}_{{\mathcal O}_K}(g)=\operatorname{Aut}_{{\mathcal O}_K}(F).$$

This removes the two additional conditions from the preceding theorem and strengthens Sarkis' conjecture on full sets of commuting formal power series. The supplied text does not state whether this strengthened conjecture has been resolved.