It's history of string theory is a bit off. The first string theory was a 26-d theory of force-like particles that are closed strings with two of the 26 dimensions being timelike. It also includes tachyons and in my opinion is the unified field. No supersymmetric particles needed.

So we should not be surprised when someone introduces an extra time dimension along with fewer than 26 dimensions.

I was suprised however that Brahman did not reference Vafa, or vice versa. The Vafa ref, the last above, does not mention hyperfine time.

Brahman talks about "Kuchar hypertime formalism".
So now that I know what you mean by hypertime I will reread your original post.