Asymptotic absence of type-III prime and twin-prime bias

Let $\delta^{+}(x)$ and $\delta^{-}(x)$ be the corresponding prime-number bias functions, and let $\delta^{+}_2(x)$ and $\delta^{-}_2(x)$ be the twin-prime bias functions measuring whether the consecutive twin-prime gap plus or minus $1$ is prime.

Type-III bias conjecture. As $x\rightarrow\infty$,

$$\delta^{+}(x)=\delta^{-}(x)=0\quad\text{and}\quad\delta^{+}_2(x)=\delta^{-}_2(x)=0.$$

The conjecture predicts that the type-III biases vanish because odd composite numbers grow faster than primes. It is conditional in the source on the infinitude of twin primes and remains open.