I have just learned <a href="https://mathoverflow.net/questions/20430/is-there-an-explicit-example-of-a-complex-number-which-is-not-a-period">here</a> that we know numbers that are not periods; is it known meanwhile that the ring of periods is not a field? I know that it is conjectured that $1/\pi$ is not a period, but the existence of a period whose inverse is not a period seems to be still open. Is this correct?

More generally: is it believed that the unit group of the ring of periods is bigger than the nonzero algebraic numbers?