Brown–Shields conjecture on cyclic functions in the Dirichlet space

Let $D_1$ denote the Dirichlet space, and let $f\in D_1$. A function is cyclic in $D_1$ if the polynomials multiplied by it are dense in $D_1$. A function $f\in D_1$ is outer if it has no nontrivial inner factor, and its set of boundary zeros is the set of points on the unit circle where its boundary values vanish. Brown–Shields conjecture. A function $f\in D_1$ is cyclic in $D_1$ if and only if it is outer and it has a set of boundary zeros of logarithmic capacity equal to zero. This conjecture seeks a complete characterization of cyclic functions in the Dirichlet space; the surrounding discussion identifies it as a question proposed by Brown and Shields, while a general characterization of cyclicity is not available.