Let X be a Stein manifold and let K be a compact subset of X. Suppose that K possesses in X a fundamental system of neighbourhoods which are Stein spaces. Then, it is a result by Rossi that such a compact subset K is holomorphically convex (see [Rossi, "Holomorphically convex sets in several complex variables," Ann. Math., 74, No. 3, 470-493 (1961)]).
Now let X be a Stein space with a finite number of singularities and let K be a connected compact subset of X containing all the singularities. Suppose that K possesses in X a fundamental system of neighbourhoods which are Stein spaces. Is it then necessarily true that K is holomorphically convex?