Wolf's sign-change conjecture for the twin-prime counting error

Let $P_2=(p,p+2)$ be the twin-prime tuple, let $\pi_2(n)$ count primes $p\leq n$ such that $p+2$ is prime, and let $C_2$ be the twin-prime constant. Define

$$d_2(n)=\pi_2(n)-C_2\operatorname{Li}_2(n),\qquad \operatorname{Li}_2(n)=\int_2^n\frac{dt}{\log^2 t}.$$

Wolf's conjecture. The number of sign changes of $d_2(n)$ for $n\in(1,T)$ is $\frac{\sqrt T}{\log T}$. This numerical prediction concerns fluctuations around the Hardy–Littlewood estimate; the paper reports substantial computational evidence, but no proof or resolution is given.