We report the first formal verification of a lock-free list, skiplist, and a skiplist-based priority queue against a strong specification in relaxed memory consistency (RMC). RMC allows relaxed behaviors in which memory accesses may be reordered with other operations, posing two significant challenges for the verification of lock-free traversals. (1) <em>Specification challenge</em>: formulating a specification that is flexible enough to capture relaxed behaviors, yet simple enough to be easily understood and used. We address this challenge by proposing the <em>per-key linearizable history specification</em> that enforces a total order of operations for each key that respects causality, rather than a total order of all operations. (2) <em>Verification challenge</em>: devising verification techniques for reasoning about the reachability of edges for traversing threads, which can read stale edges due to relaxed behaviors. We address this challenge by introducing the <em>shadowed-by relation</em> that formalizes the notion of outdated edges. This relation enables us to establish a total order of edges and thus their associated operations for each key, required to satisfy the strong specification. All our proofs are mechanized on the iRC11 relaxed memory separation logic, built on the Iris framework in Rocq.