Conditional hardness for [US:US:GM] remains unresolved

Establish a conditional hardness result for computing X = AB in the supported low-bandwidth model when A and B are uniformly sparse matrices (each row and column has at most d nonzeros) and the set of required outputs is general (no sparsity constraint), i.e., the case [US:US:GM].

Background

The authors summarize upper bounds and conditional hardness results across various sparsity combinations. While they obtain hardness for several cases and upper bounds up to [US:AS:GM], they explicitly note a gap: no conditional hardness result is provided for [US:US:GM].

Filling this gap would help delineate the limits of fast algorithms when both inputs are uniformly sparse but the outputs of interest are not, potentially linking this case to fundamental communication primitives similar to those used in other hardness arguments.