The conjecture that the stated monodromy condition characterizes infinite distortion

Let $X\to B$ be an almost Postnikov pair with $\pi_1X=\pi_1B=\Gamma$, monodromy representation $\rho$, and Euler class $\operatorname{eu}$. Suppose that for some $\gamma\in\Gamma$, the pullback $\gamma^*\operatorname{eu}$ is nontrivial when restricted to a direct summand $\mathbb{Q}\cong L\subseteq M_{\gamma^*\rho}$ on which the action of $\gamma$ is trivial. Infinite-distortion characterization conjecture. The condition stated above is equivalent to the presence of infinite distortion in $\pi_n(X)$. The source notes that this condition is necessary in the case of trivial $\rho$ and conjectures that the equivalence holds in general; however, it also supplies a counterexample to the conjecture, so the claim is refuted.