Derivative-generation conjecture for invariant subspaces of the weighted Bergman space

Let $A^2_1$ be the weighted Bergman space of holomorphic functions on the unit disk with norm

$$\|f\|_{A^2_1}=\left(\int_{\mathbb{D}}|f(z)|^2(1-|z|)\,|dz|^2\right)^{1/2}<\infty.$$

For $H\in A^2_1$, write

$$[H]=\overline{\{Hp:p\text{ polynomial}\}},$$

the smallest closed subspace of $A^2_1$ containing $H$ and invariant under multiplication by $z$. An inner function is a bounded holomorphic function on the unit disk whose radial boundary values have modulus one almost everywhere. Derivative-generation conjecture. If an invariant subspace of $A^2_1$ can be generated by a single function, then it can be generated by the derivative of an essentially unique inner function. Shimorin's result identifies the closure of zero-based subspaces with singly generated invariant subspaces, while Kraus's theorem identifies critical sets of Blaschke products with those of $H^2$ functions. The conjecture seeks the corresponding derivative description and uniqueness, modulo the natural equivalence of inner functions by disk automorphisms; its resolution is not established in the supplied text.