Erdős Problem 1212

Let $G$ be the graph with vertex set those pairs $(x,y)\in \mathbb{N}^2$ with $\mathrm{gcd}(x,y)=1$, in which we join two vertices if the differ in only one coordinate, and there by $\pm 1$.

Is there a path going to infinity on $G$, say $P$, such that for all $(x,y)\in P$ both $\min(x,y)>1$ and at least one of $x$ or $y$ is composite?

The weaker version (only $\min(x,y) > 1$) was solved by C. Stewart via the prime-pair path $(p_k, p_{k+1}) \to (p_{k+1}, p_{k+2})$, as recounted in [Er80]; the compositeness condition forbids those anchors and the question is open.