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It seems to me that the answer to your question is no, because of the following example (which came to my mind after reading Artie Prendergast-Smith's comment).

Consider a pencil $\lambda Q_1 + \mu Q_2$ of quartic surfaces in $\mathbb{P}^3$, and let $Z$ be its base locus, that in general will be a smooth curve of degree $16$. Blowing-up $Z$, we obtain a smooth rationally connected $3$-fold $X$ together with a map $\pi \colon X \to \mathbb{P}^1$, which gives to $X$ the structure of a fibration in $K3$ surfaces. If $F$ is the class of a fibre of $\pi$, the formula for the canonical class of a blow-up yields

$K_S=-F$. K_X=-F$. So for every surface$S \subset X$one has$(-K_S)^2 (-K_X)^2 \cdot S=0$. This can be obviously generalized in any dimension, by considering a pencil of hypersurfaces of degree$n+1$in$\mathbb{P}^n$and blowing-up the corresponding base locus. In this way one obtain a smooth rationally connected$n$-fold$X$with a fibration$\pi \colon X \to \mathbb{P}^1$in Calabi-Yau varieties, and the anticanonical divisor of$X$coincides with a fibre of$\pi$. 4 added 57 characters in body It seems to me that the answer to your question is no, because of the following example (which came to my mind after reading Artie Pradengast-Smith's Prendergast-Smith's comment). Consider a pencil$\lambda Q_1 + \mu Q_2$of quartic surfaces in$\mathbb{P}^3$, and let$Z$be its base locus, that in general will be a smooth curve of degree$16$. Blowing-up$Z$, we obtain a smooth rationally connected$3$-fold$X$together with a map$\pi \colon X \to \mathbb{P}^1$, which gives to$X$the structure of a fibration in$K3$surfaces. If$F$is the class of a fibre of$\pi$, the formula for the canonical class of a blow-up yields$K_S=-F$. So for every effective divisor surface$S \subset X$one has$(-K_S)^2 \cdot S=0$. This can be obviously generalized in any dimension, by considering pencils a pencil of hypersurfaces of degree$n+1$in$\mathbb{P}^n$and blowing-up the corresponding base locus. In this way one obtains obtain a smooth rationally connected$n$-folds n$-fold $X$ with a fibration $\pi \colon X \to \mathbb{P}^1$ in Calabi-Yau varieties, and the anticanonical divisor of $X$ coincides with a fibre .of $\pi$.

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It seems to me that the answer to your first question is no, because of the following example (which came to my mind after reading Artie Pradengast-Smith's comment).

Consider a pencil $\lambda Q_1 + \mu Q_2$ of quartic surfaces in $\mathbb{P}^3$, and let $Z$ be its base locus, that in general will be a smooth curve of degree $16$. Blowing-up $Z$, we obtain a rationally connected $3$-fold $X$ together with a map $\pi \colon X \to \mathbb{P}^1$, which gives to $X$ the structure of a fibration in $K3$ surfaces. If $F$ is the class of a fibre of $\pi$, the formula for the canonical class of a blow-up yields

$K_S=-F$.

So for every effective divisor $S \subset X$ one has $(-K_S)^2 S=0$.

This can be obviously generalized in any dimension, by considering pencils of hypersurfaces of degree $n+1$ in $\mathbb{P}^n$ and blowing-up the corresponding base locus. In this way one obtains smooth rationally connected varieties $n$-folds with a fibration in Calabi-Yau manifoldsvarieties, and the anticanonical divisor coincides with a fibre.

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