Achieving full sharing of level-ℓ blocks across all dimensions in laser-method analyses

Develop a laser-method-based procedure for powers of the Coppersmith–Winograd tensor CW_q that allows different remaining level-ℓ subtensors to share level-ℓ variable blocks in all three dimensions (X, Y, Z), while requiring only that each level-1 variable block belongs to a unique remaining level-ℓ subtensor.

Background

Recent improvements (Duan–Wu–Zhou; Vassilevska Williams–Xu–Xu–Zhou) introduced asymmetry to mitigate combination loss in recursive laser-method analyses, first permitting sharing in Z and later finer-grained sharing. The authors further advance this by enabling sharing in Y and Z while retaining uniqueness of level-ℓ X-blocks.

They state the natural end goal of the line of work: sharing level-ℓ blocks in all three dimensions with only level-1 uniqueness constraints. They explicitly note that achieving this goal is presently unclear, highlighting a central methodological open direction.