Moment Generating Function of a Truncated Sub-Gaussian

Question

This is a follow-up to a simpler question. Say that random variable $X$ is sub-gaussian with unit proxy variance, i.e., $P(|X|\ge t)\le 2 \exp(-t^/2)$. The moment generating function of $X$ is bounded above by $\exp(t^2)$ (which is the MGF of a standard gaussian). Let $Y=X\mathbb{I}(|X|\ge a)$. What is the MGF $E[\exp(tY)]$ of $Y$? Intuitively, it should be upper-bounded by the answer I linked to.