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

1. What is the best estimate known for this $c$?

2. Is $\frac12-c=o(1)$ believed achievable with current techniques?

3. Is the technique applicable to union of disjoint intervals without testing them separately?