It is known that recognizing a unit disk graph is NP-hard [1].

However, the paper does not mention how hard is the realization problem.

I have looked up several references [2][3][4]. None of the papers answer whether the following problem is NP-hard:

Given a unit disk graph $ G = (V,E)$ , find a configuration of a set $ \mathcal{D}$ of disks, such that the intersection graph $ G(\mathcal{D})$ of $ \mathcal{D}$ is isomorphic to $ G$ .

The difference between this problem and the recognition problem is that the input of this problem is guaranteed to be a unit disk.

Is there any study that shows the complexity of the above problem? I expect it to be NP-hard, but I am yet to find a full proof.