Erdős Problem #1212 — An Infinite Coprime Lattice Path with Nonprime Coordinates

Consider the graph with vertex set consisting of pairs $(x,y)\in\mathbb N^2$ such that

$$1<x,\qquad 1<y,\qquad \gcd(x,y)=1,$$

and at least one of $x$ or $y$ is nonprime. Two vertices $p=(p_1,p_2)$ and $q=(q_1,q_2)$ are adjacent exactly when

$$\bigl(p_1=q_1\ \land\ (p_2=q_2+1\ \lor\ q_2=p_2+1)\bigr) \quad\lor\quad \bigl(p_2=q_2\ \land\ (p_1=q_1+1\ \lor\ q_1=p_1+1)\bigr).$$

Does there exist an injective sequence $f:\mathbb N\to\mathbb N^2$ such that every consecutive pair $f(n),f(n+1)$ is adjacent, every $f(n)$ is a vertex of this graph, and

$$\lim_{n\to\infty}\bigl(f(n)_1+f(n)_2\bigr)=\infty?$$