Balanced versus lopsided All-Edges Sparse Triangle

Determine whether the balanced version of All-Edges Sparse Triangle requires near-$m^{4/3}$ time even though the lopsided version admits faster algorithms; equivalently, establish whether a faster algorithm for lopsided instances can coexist with the conjectured hardness of balanced instances.

Background

The paper’s main algorithm exploits a tripartite graph with two large parts and a substantially smaller middle part. It does not improve the balanced case, which is the target of several lower-bound reductions and is conjectured to require $m^{4/3-o(1)}$ time. The authors explicitly ask whether the observed separation between the two parameter regimes is genuine.