Linear-time implication between transposition and multiplication

Determine whether the inability to perform binary matrix transposition in linear time in the multitape Turing model implies that integer multiplication cannot be performed in linear time. Concretely, prove or refute that if Tcost_T(m) ≠ O(m) for all binary transposition machines T, then Mcost_M(m) ≠ O(m) for all multiplication machines M.

Background

The paper proves a conditional result: if no transposition machine achieves O(m f(m) log* m), then no multiplication machine achieves O(m f(m)). The authors would like a stronger, direct statement that non-linearity for transposition forces non-linearity for multiplication, without the log* factor. Their current methods do not suffice, leaving this implication open.