Entropy of logarithmic modes

Let $(M,g)$ be a compact, boundaryless, Riemannian manifold whose geodesic flow on its unit sphere bundle is Anosov. Consider the (semiclassical) Laplace-Beltrami operator on $M$. Let $\epsilon >0$. We study the semiclassical measures $\mu_{sc}$ of quasimodes spectrally supported in intervals of width $\epsilon \frac{h}{|\log h|}$, a critical-type regime when considering ``delocalization". We derive a lower bound for the Kolmogorov-Sinai entropy of $\mu_{sc}$ that depends explicitly on $\epsilon$, in the spirit of that given by Ananthamaran-Koch-Nonnenmacher.