Does there exist a computable number that is normal in all bases?

Following up on this exchange with Marty Cohen...

Almost all numbers are normal in all bases (absolutely normal), but there are only a countable number of computable numbers, so it is plausible that none of them are absolutely normal. Now I don't expect to be able to prove this since it would imply $\pi$, $\sqrt{2}$, etc. are not absolutely normal. Also I don't expect to be able to find a particular computable number that is normal in all bases, since Marty states none are known. But is it possible to show non-constructively that there is some computable number which is absolutely normal?

Below are a couple of papers for what you want. For more, google computable absolutely normal.

Verónica Becher and Santiago Figueira, An example of a computable absolutely normal number, Theoretical Computer Science 270 #1-2 (6 January 2002), 947-958. [Another copy here.]

Verónica Becher, Pablo Ariel Heiber, and Theodore A. Slaman, A computable absolutely normal Liouville number, Mathematics of Computation 84 #296 (November 2015), 2939-2952.