Let $X$ be a compact complex manifold with canonical bundle $K_X$. Assume the Kodaira dimension $\kappa(X)$ is positive (but *not* maximal, i.e., $X$ is *not* of general type). Let $\varphi_m : X \dashrightarrow Y_m \subset \mathbb{P}^{N_m}$ denote the $m$th pluricanonical map given by the sections of the $m$th tensor power $K_M^{\otimes m}$.

Suppose $X$ does not contain any rational curves. Does $Y_m$ contain any rational curves?

In more detail, if $K_X$ is semi-ample, the base of the Iitaka map $\varphi : X \to X_{\text{can}}$ given by the linear system $|K_X^{\otimes \ell} |$ for $\ell>0$ sufficiently large has ample canonical bundle (I've seen this stated, but am not certain about $K_{X_{\text{can}}}$ being ample, for general $X$ with $K_X$ semi-ample). In any case, if $K_{X_{\text{can}}}$ is ample, then Mori's newness result says that $X_{\text{can}}$ has no rational curves.

I'm wondering if this type of phenomenon occurs for the pluricanonical maps, in general, *eventually*. That is,

Suppose $X$ does not contain any rational curves. For $m>0$ sufficiently large, do the base spaces $Y_m$ of the pluricanonical maps have rational curves?