If contention is expected, would it be better to first perform a relaxed read before the exchange? For example:

  auto lock() noexcept -> void {
      auto backoff = 1;
      do {
          while (locked_.load(std::memory_order_relaxed)) {
              for (auto i = 0; i < backoff; ++i) _mm_pause();
              backoff = backoff < 64 ? backoff << 1 : 64;
          }
      } while (locked_.exchange(true, std::memory_order_acquire);
  }