abc conjecture for a Galois number field

Let $K$ be a Galois number field and let $\alpha,\beta\in K$. Using the height $H$ and conductor $N$ defined in the source by

$$H(a_1,a_2,a_3)=\prod_v\max(\lVert a_1\rVert_v,\lVert a_2\rVert_v,\lVert a_3\rVert_v),$$

and

$$N(a_1,a_2,a_3)=\prod_{\mathfrak p\in I}\operatorname{Nm}_{K/\mathbb{Q}}(\mathfrak p)^{1/[K:\mathbb{Q}]},$$

where $I$ is the finite set of normalized finite places at which the three normalized absolute values are not all $1$, the ideal notation $(\alpha,\beta,\alpha+\beta)$ denotes the ideal generated by these elements.

The abc conjecture for a Galois number field. For every $\epsilon>0$,

$$H(\alpha,\beta,\alpha+\beta)\ll_{\epsilon,K}N\left((\alpha,\beta,\alpha+\beta)\right)^{1+\epsilon}.$$

The source states that this conjecture is equivalent to Elkies' version of Vojta's abc conjecture for number fields. The supplied text gives no resolution status.