How Close Can Numbers with Bounded Prime Factors Be?

Question

How close can numbers with bounded prime factors be? Specifically, let $\text{gpf}(n)$ be the greatest prime factor that divides $n$. Is there any way to analyze the function $$G(x)=\lim_{y\to\infty}\inf\{|n-m|\; :\; n\neq m,\; \text{gpf}(n),\text{gpf}(m)\le x,\; n,m\ge y\}?$$ I am only concerned with the limit because clearly we could achieve a minimum of one by simply choosing $n,m\le x$ otherwise. My guess is that this limit goes to infinity, in which case I am more concerned with how fast the quantity in the limit grows. We may also assume that $n$ and $m$ are odd if it helps.