The normal-or-analytic conjecture for subnormal Toeplitz operators

Let $T_\phi$ be a scalar Toeplitz operator on the Hardy space whose self-commutator $[T_\phi^*,T_\phi]$ has finite rank. Normal-or-analytic conjecture. If $T_\phi$ is subnormal, then $T_\phi$ is normal or analytic. This remains open even for scalar-valued Toeplitz operators and would characterize subnormal Toeplitz operators with finite-rank self-commutators as normal or analytic.