Kirby’s Whitehead-double concordance conjecture

Let $K$) be a knot, let $D(K)$ denote its positively-clasped Whitehead double, and let $[K]$ and $[D(K)]$ denote their elements in the knot concordance group $\mathcal{C}$. Kirby’s Whitehead-double conjecture. As elements of the knot concordance group $\mathcal{C}$, the equality $[D(K)]=0$ holds if and only if $[K]=0$. Since $D(K)$ has trivial Alexander polynomial, it is topologically slice; the conjecture asks whether its smooth concordance class can vanish only when that of $K$ does. Hedden and Kirk formulated a related linear-independence conjecture for the map sending $[K]$ to $[D(K)]$, proving it for the family $\mathcal{T}_{2,2^n-1}$ with $n\geq2$.