Is every attached analytic disc image a multiplier variety?

Determine whether, for any injective analytic map f: D → B_d that extends smoothly up to the boundary and satisfies ||f(x)|| = 1 if and only if |x| = 1, the image V = f(D) is necessarily a multiplier variety (i.e., a joint zero set of functions from the multiplier algebra M_d).

Background

The notion of an analytic disc attached to the unit sphere is defined via an embedding map f: D → B_d that extends up to the boundary and meets the unit sphere transversally. In the thesis, such discs are treated as varieties to enable the associated Hilbert function spaces and multiplier algebras to be defined.

It remains unresolved whether the geometric conditions on f alone guarantee that its image V = f(D) is a multiplier variety without assuming this property by definition.