Erdős–Turán conjecture on largest prime factors of consecutive integers

Let $P^+(n)$ denote the largest prime factor of $n$, with $P^+(1)=1$. Erdős–Turán conjecture. As $x$ tends to infinity,

$$\#\{n\leq x:P^+(n)<P^+(n+1)\}\sim \frac{1}{2}x.$$

This conjecture concerns the expected symmetry between the largest prime factors of consecutive integers. The paper proves only a lower bound of $0.280$ for the corresponding asymptotic density, so the asserted density $1/2$ remains open.