Testing whether a bounded-degree $n$-vertex graph is bipartite or far from bipartite requires $\widetilde\Theta(\sqrt n)$ queries (Goldreich--Ron, Combinatorica'99, Algorithmica'02). An interactive proof of proximity (IPP) is a hybrid model of property testing and interactive proofs: the verifier faces the same task with the same query access, but it now interacts with an untrusted prover that sees the whole graph. Rothblum, Vadhan, and Wigderson (STOC'13) introduced this model and gave a one-round private-coin IPP for bipartiteness on well-mixing graphs with query and communication costs both $O(\log n)$ for fixed parameters, an exponential improvement over plain testing. The mixing assumption was used in their soundness analysis, and they asked whether it could be removed. We give a slightly modified version of the original RVW protocol and prove its soundness without the mixing promise. In the RVW protocol, soundness follows by bounding the probability of correctly guessing the parity of a hidden random walk from its start and endpoint. The best guess is wrong with probability equal to the overlap between the endpoint probabilities for even and odd numbers of edge traversals, and mixing gives one way to bound this overlap. We prove an overlap theorem that requires no mixing, building on the recent semidefinite programming (SDP) analysis of the Goldreich--Ron tester by Fei and Rubinfeld (2026).