In https://arxiv.org/abs/1009.3956 is it shown there is a $c>0$ such that $\pi(x)\bmod2$ can be computed in $o(x^{\frac12})$ time (more precisely number of primes $\bmod 2$ for an interval of length at most $O(x^{\frac12+c})$ in $[x,2x]$ can be computed in time $O(x^{\frac12-c})$)?

If so is there a good estimate known for $c$ and is $\frac12-c=o(1)$ believed achievable?