Does same self-crossing type imply multiplier algebra isomorphism?

Determine whether, for analytic discs V = f(D) and W = g(D) attached to the unit sphere whose embedding maps f and g have the same self-crossing type on the boundary (i.e., there exists a disc automorphism µ such that f(ξ) = f(ζ) if and only if g(µ(ξ)) = g(µ(ζ)) for all ξ, ζ ∈ T), the multiplier algebras M_f and M_g (Mult(H_f) and Mult(H_g)) are algebraically isomorphic.

Background

The thesis proves that an algebraic isomorphism of multiplier algebras for attached analytic discs forces a correspondence of boundary self-crossings up to a disc automorphism, giving a necessary geometric condition for isomorphism.

Whether this necessary condition is also sufficient remains unresolved, leaving open the classification of multiplier algebras in terms of boundary self-crossing type.