Allen's conjecture on sums of torus knots concordant to L-space knots

Let a connected sum be formed from possibly several torus knots, and let an $L$-space knot mean a knot admitting a positive Dehn surgery yielding an $L$-space. Two knots are concordant if they cobound a smoothly embedded annulus in $S^3\times[0,1]$.

Allen's conjecture. If a connected sum of possibly several torus knots is concordant to an $L$-space knot, then it is concordant to a positive torus knot.

This conjecture extends results of Livingston and Allen on concordance obstructions for connected sums of torus knots. The paper proves the conjecture for alternating torus knots, while the general case remains open.