The injectivity conjecture for positive Whitehead doubles of slice disks

Let $K$ be a knot, and let $D_1$ and $D_2$ be slice disks for $K$ in $B^4$. Their positive Whitehead doubles are the slice disks obtained by applying the positive Whitehead satellite construction to $D_1$ and $D_2$. Slice-disk Whitehead-double conjecture. The disks $D_1$ and $D_2$ are smoothly isotopic if and only if their positive Whitehead doubles are smoothly isotopic. The disks are topologically isotopic under the relevant cyclic-complement hypotheses, so the conjecture asks whether the positive Whitehead-double operation detects smooth isotopy classes of slice disks; its status is not resolved in the source.