Conjecture on friable neighboring integers

For $u,v\geq 1$, let

$$\Psi(x;u,v):= \\{1\leq n\leq x: P^+(n)\leq x^{1/u}\text{ and } P^+(n+a)\leq x^{1/v}\\}.$$

Here $P^+(m)$ denotes the largest prime factor of $m$, and $\varrho$ is the Dickman function. Neighboring friable integers conjecture. For all $u,v\geq 1$,

$$\Psi(x;u,v)\sim \varrho(u)\varrho(v)x$$

when $x\to\infty$. This predicts asymptotic independence of the friability conditions on $n$ and $n+a$; the source presents it as a heuristic conjecture, with no resolution supplied here.