A condition for asymptotic finite-dimensionality of an operator semigroup

Let $X$ be a Banach space and let $T\colon X\to X$ be a power bounded linear operator. Put $X_0=\{x\in X\mid T^nx\to0\}$. Assume given a compact set $K\subset X$ such that $\liminf_{n\to\infty}\rho\{T^nx,K\}\le\eta<1$ for every $x\in X$, $\|x\|\le1$. If $\eta<\frac12$, then $\operatorname{codim}X_0<\infty$. This is true in $X$ reflexive for $\eta\in[\frac12,1)$, but fails in the general case.