Asymptotic cardinality conjecture for universal sets in the Gaussian integers

Let an $n$-universal set in $d4e[ i ]$ be a finite set such that every polynomial of degree at most $n$ taking integer values on that set takes integer values at every point of $d4e[ i ]$. Let $m_n$ denote the minimal cardinality of an $n$-universal set.

Asymptotic cardinality conjecture. The minimal cardinality satisfies

$$m_n=\frac{\pi}{2}n+o(n),$$

and an asymptotically sharp example is realized by the set of integer points inside a circle of radius

$$\sqrt{n/2}+o(\sqrt n).$$

This conjecture generalizes the two preceding theorems, which provide lower and upper bounds for the size of universal sets. Its resolution would determine the asymptotically minimal size and the geometric form of an asymptotically optimal example.