Multiplier biholomorphism implies algebraic isomorphism? (non-discrete varieties)

Determine whether, for analytic varieties V and W in the unit ball B_d that are not discrete, the existence of a multiplier biholomorphism between V and W implies that their multiplier algebras M_V and M_W are algebraically isomorphic.

Background

The thesis surveys known implications between geometric relationships of varieties in the unit ball and algebraic relationships of their associated multiplier algebras. For homogeneous varieties in finite dimensions, algebraic isomorphism of multiplier algebras is equivalent to multiplier biholomorphism, and for irreducible varieties an algebraic isomorphism implies multiplier biholomorphism.

However, the converse direction—whether multiplier biholomorphism necessitates algebraic isomorphism—remains unsettled outside special cases. It is known to fail for discrete varieties, leaving open the question for non-discrete classes of varieties.