  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 $\epsilon$ average swap regret against an adversary in any
extensive-form game within $m^{\tilde{\mathcal O}(1/\epsilon)}$ rounds, where
$m$ is the number of nodes in the game tree. However, the question of whether a
$\mathrm{poly}(m, 1/\epsilon)$-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 $\epsilon$
against an oblivious adversary in general extensive-form games requires at
least $\mathrm{exp}\left(\Omega\left(\min\left\{m^{1/14},
\epsilon^{-1/6}\right\}\right)\right)$ rounds.
