The Li-Li conjecture, also called the undirected multiple-unicast conjecture, asserts that network coding does not improve the maximum common transmission rate over fractional routing in an undirected network with several independently specified source-destination pairs. Fractional routing divides each message among graph paths, whereas network coding allows intermediate vertices to transmit functions of received messages.
Braverman and He (2026) reported an explicit counterexample with 182 vertices, 735 edges, and 157 source-destination pairs. A linear
network code over the finite field 
achieves rate at least 1 per pair, whereas the fractional
routing rate is at most 
.
Independently, Zhang et al. (2026) reported a counterexample admitting binary linear network coding and a family with maximum vertex degree 3 for which the coding advantage grows polylogarithmically with network size. Their family works over every finite field. Zhang et al. (2026) provide a Lean development, which has not been independently checked here.
Braverman and He (2026) report that GPT-6 contributed the coding architecture underlying their construction. They developed the causal orientation and verified the finite routing certificate.
