Is it a difficult problem to determine if a knot is of the form $K \# K$?

Question

I know that determining whether a knot is prime is 'difficult' (of course this has no rigorous definition, and I am open to people using whatever definition allows them to give a definite answer to this question). I have the sense that many basic questions like this around knots are difficult, so I suspect that the question of whether a given knot $L$ is equal to the connected sum of a knot $K$ with itself will also be difficult. However I don't know how to search for references for this. This is motivated by an answer I wrote showing that this question can be embedded in the question of whether an element of a GCD is square.