The inverse Galois conjecture for semi-topological coverings

Let $X$ be a space and let $Y \stackrel{p}{\rightarrow}X$ be a finite connected Galois covering of $X$. A Weierstrass polynomial $f$ on $X$ is a polynomial whose splitting covering is defined as $E_f \stackrel{q}{\rightarrow}X$.

Semi-topological inverse Galois conjecture. There exists a Weierstrass polynomial $f$ on $X$ such that the splitting covering $E_f \stackrel{q}{\rightarrow}X$ is equivalent to the covering $Y \stackrel{p}{\rightarrow}X$.

The conjecture asks whether every finite connected Galois covering can be realized as the splitting covering of a Weierstrass polynomial. The source explicitly states that this problem had not been solved at the time of writing.