# Is realization of unit disk graphs hard?

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.