Is it the case that, for any given knot $K$, there exists some graph $G$ whose every embedding into $\mathbb{R}^3$ (or into $\mathbb{S}^3$) contains a cycle that realizes $K$?

I know the famous Conway-Gordon result that the complete graph
on seven vertices $K_7$ is intrinsically knotted in that every
embedding contains a knotted cycle.
And Joel Foisy proved that $K_{3,3,1,1}$ is also intrinsically knotted
(*J. Graph Theory*, 2002, ACM link).

Instead, I am asking a Ramsey-like question: Whether it's known that, for every particular knot, there exists some graph for which that particular knot is unavoidable.