Is the invariant subspace problem open for invertible maps?

Let $T:H\to H$ be a bounded linear operator with bounded inverse on the separable complex Hilbert space. Does T preserve a closed proper non-trival invariant subspace?

I'm aware the question is (famously) open for bounded linear maps, and of partial results, but no survey (or Tao's blog, etc) seem to address the invertible case.

If it is open, does a positive or negative answer imply the answer in the non-invertible case?

Let $T:H\to H$ be a bounded linear operator with bounded inverse on the separable complex Hilbert space. Does T preserve a closed proper non-trival invariant subspace?

I'm aware the question is (famously) open for bounded linear maps, and of partial results, but no survey (or Tao's blog, etc) seem to address the invertible case.

If it is open, does a positive or negative answer imply the answer in the non-invertible case?