Log Rank Conjecture Collaborative Approach
it seems researchers might be cautious/ unlikely to conjecture/ speculate solution to this problem would likely/ definitely open up other areas (eg one you list of matrix rigidity is provably nearly equivalent to the extremely hard challenge of proving nontrivial lower bounds on eg circuits etc) but it definitely provably touches many areas. here is a nice recent survey (Jan 2015) by Lovett that lists "many connections" sec 6 along with two other refs that show crosscutting connections/ formulations


Recent advances on the log-rank conjecture in
communication complexity / Lovett


6.1 Randomized protocols vs approximate rank
6.2 Quantum protocols for low-rank matrices
6.3 The structure of low-rank sparse matrices, and matrix rigidity

En Route to the log-rank Conjecture:
New Reductions and Equivalent Formulations / Gavinsky, Lovett
On the "log rank"-Conjecture in Communication Complexity / Raz, Spieker