Gordon's order-two concordance conjecture

Let $K\subseteq S^3$ be a knot, and write $K\#K$ for its connected sum with itself. A knot is negative amphichiral if it is isotopic to $r\overline{K}$, where $r$ denotes reversal and $\overline{K}$ denotes the mirror. Gordon's conjecture. If $K\#K$ is slice, then $K$ is smoothly concordant to a negative amphichiral knot. This asks whether every element of order at most two in the smooth knot concordance group is represented, up to smooth concordance, by a negative amphichiral knot.