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.