Erdős–Pomerance conjecture on independent largest prime factors

Let $P^+(n)$ denote the largest prime factor of $n$. For $a,b\in[0,1]$, define

$$B(x;a,b):=\#\{n\leq x:P^+(n)\leq x^a,\ P^+(n+1)\leq x^b\}.$$

Let $\rho(t)$ denote the Dickman–de Bruijn function. Erdős–Pomerance conjecture. For every $a,b\in[0,1]$,

$$\lim_{x\rightarrow\infty}x^{-1}B(x;a,b)=\rho\left(\frac{1}{a}\right)\rho\left(\frac{1}{b}\right).$$

This formalizes the conjectured independence of the largest prime factors of consecutive integers and implies the Erdős–Turán conjecture. Its resolution status is not specified in the supplied text, and it remains open here.