Let $K_0$ and $K_1$ be two knots in $S^3$. We say $K_0$ and $K_1$ are concordant if there exists a smoothly embedded annulus $A \subset S^3 \times [0,1] $ such that $\partial A = -(K_0) \cup K_1$.
Given two non-trivial concordant knots $K_0$ and $K_1$, assume that one of them is hyperbolic, say $K_0$. Is it possible to show that $K_1$ must be hyperbolic?
We may ask the similar question by changing the "hyperbolic" notion with "amphichiral". In other words, does the knot concordance preserve the hyperbolicity or amphichirality of the knot?