Ivrii's conjecture on singly generated invariant subspaces of the Bergman space

Let $A^2_1$ be the Bergman space

$$A^2_1=\left\{\varphi:\mathbb D\to\mathbb C\text{ holomorphic}:\iint_{\mathbb D}(1-|z|^2)|\varphi(z)|^2\,dx\,dy<\infty\right\}.$$

A closed subspace of $A^2_1$ is singly generated if it is the minimal closed invariant subspace containing one function, where invariance is with respect to multiplication by $z$. An inner function is a bounded holomorphic function on the unit disk with unimodular boundary values almost everywhere. Ivrii's conjecture. Any singly generated subspace of $A^2_1$ can be generated by the derivative of an inner function. The inner function is uniquely determined up to postcomposition with a unit disk automorphism. Equivalently,

$$\text{Inner functions}/\operatorname{Aut}(\mathbb D)=\{\text{singly generated invariant subspaces of }A^2_1\}.$$

This conjecture connects inner functions and invariant-subspace theory in Bergman spaces, extending the correspondence between derivatives of maximal Blaschke products and zero-based invariant subspaces. The source presents it as an open conjecture motivated by asymptotic spectral synthesis and Shimorin's approximation theorem.