Heterochromatic two-arm probabilities for metric graph Gaussian free fields

For the Gaussian free field on the metric graph of $\mathbb{Z}^d$ ($d\ge 3$), we consider the heterochromatic two-arm probability, i.e., the probability that two points $v$ and $v'$ are contained in distinct clusters of opposite signs with diameters at least $N$. For all $d\ge 3$ except the critical dimension $d_c=6$, we prove that this probability is asymptotically proportional to $N^{-[(\frac{d}{2}+1)\land 4]}$. Furthermore, we prove that conditioned on this two-arm event, the volume growth of each involved cluster is comparable to that of a typical (unconditioned) cluster; precisely, each cluster has a volume of order $M^{(\frac{d}{2}+1)\land 4}$ within a box of size $M$.