No exponentially convergent spectral-ratio families in the P≤ cases

Let $h$ be a pseudo-Anosov mapping class and let $\operatorname{\omega}(h)$ denote its spectral ratio. For a surface and an infinite family of pseudo-Anosov mapping classes, a P$_{\leq}$ case means that $\operatorname{\omega}(h)\leq 1+1/\operatorname{poly}(|h|)$, while an E case means that $\operatorname{\omega}(h)\leq 1+1/\exp(|h|)$. P$_{\leq}$ versus E conjecture. In all P$_{\leq}$ cases, there is no family of pseudo-Anosov mapping classes whose spectral ratios converge to one exponentially; equivalently, none of the P$_{\leq}$ cases are E cases. The claim distinguishes polynomially slow convergence of the spectral ratio from exponential convergence in the surface-by-surface classification. The supplied text gives no resolution, so the assertion remains open.