Nonconcordance conjecture for iterated Whitehead doubles of the trefoil

Let $T_{2,3}$ be the $(2,3)$ torus knot, and let $D(K)$ denote its positively-clasped untwisted Whitehead double. Two knots are smoothly concordant when their connected sum with one orientation-reversed mirror is smoothly slice. Nonconcordance conjecture for iterated Whitehead doubles of the trefoil. The knots

$$D(T_{2,3})\quad\text{and}\quad D^2(T_{2,3})$$

are not smoothly concordant. All concordance invariants discussed in the source take the same values on these knots, so their smooth concordance remains mysterious and the conjecture is open.