Is the problem Coiling Rope in a Box decidable? To be specific, is this decidable?

Given $L > 0$ and $r \in (0,\frac{1}{2})$, both rational, can a rope of length $L$ and radius $r$ fit into a unit-cube box?

See the earlier MO question linked above for the problem definition.

It seems one would have to represent all possible rope curves with a finite set of parameters, and then use Tarski's quantifier elimination. But perhaps there are other routes to determining decidability.

Is the problem Coiling Rope in a Box decidable? To be specific, is this decidable?

Given $L > 0$ and $r \in (0,\frac{1}{2})$, both rational, can a rope of length $L$ and radius $r$ fit into a unit-cube box?

See the earlier MO question linked above for the problem definition.

It seems one would have to represent all possible rope curves with a finite set of parameters, and then use Tarski's quantifier elimination. But perhaps there are other routes to determining decidability.