Triple-iterated logarithm implication from transposition to multiplication

Establish that, in the multitape Turing model, a lower bound Tcost(m) = ω(m log log log m) for binary matrix transposition implies a corresponding lower bound Mcost(m) = ω(m log log log m) for integer multiplication of m-bit integers.

Background

The authors present several implications showing that superlinear lower bounds for transposition yield superlinear bounds for multiplication. They prove a sequence of weaker implications (e.g., for O(m log^{∘ℓ} m) regimes), but note a gap at the triple-iterated logarithm threshold: they cannot currently show that ω(m log log log m) for transposition forces the same bound for multiplication. Resolving this would sharpen the correspondence between the two problems.