Recent simultaneous works by Peng and Rubinstein [2024] and Dagan et al. [2024] have demonstrated the existence of a no-swap-regret learning algorithm that can reach $ε$ average swap regret against an adversary in any extensive-form game within $m^{\tilde{\mathcal O}(1/ε)}$ rounds, where $m$ is the number of nodes in the game tree. However, the question of whether a $\mathrm{poly}(m, 1/ε)$-round algorithm could exist remained open. In this paper, we show a lower bound that precludes the existence of such an algorithm. In particular, we show that achieving average swap regret $ε$ against an oblivious adversary in general extensive-form games requires at least $\mathrm{exp}\left(Ω\left(\min\left\{m^{1/14}, ε^{-1/6}\right\}\right)\right)$ rounds.