Asymptotic absence of type-II twin-prime bias

For residue classes $a_i$ and $a_j$ modulo $4$, let $\delta_2(x;4;a_i\mid a_j)$ denote the type-II twin-prime bias function comparing the relevant twin-prime counts in the two classes.

Type-II bias conjecture. For any two residue classes $a_i$ and $a_j$,

$$\delta_2(x;4;a_i\mid a_j)=0.5\quad\text{as }x\rightarrow\infty.$$

The conjecture predicts that the observed type-II bias eventually disappears, assuming infinitely many twin-prime pairs. Its status is open because twin-prime infinitude and the asserted limiting behavior are unproved.