32. From the generalization of Cauchy's inequality, we have
$$\left(a_{k}^{3}+1\right)\left(a_{k}^{3}+1\right)\left(a_{k+1}^{3}+1\right) \geqslant\left(a_{k}^{2} a_{k+1}+1\right)^{3}, \quad k=1,2, \cdots, n,$$
where $a_{n+1}=a_{1}$.

Multiplying them together, we get
$$\prod_{k=1}^{n}\left(a_{k}^{3}+1\right)^{3} \geqslant \prod_{k=1}^{n}\left(a_{k}^{2} a_{k+1}+1\right)^{3}$$

which simplifies to
$$\left(a_{1}^{3}+1\right)\left(a_{2}^{3}+1\right) \cdots\left(a_{n}^{3}+1\right) \geqslant\left(a_{1}^{2} a_{2}+1\right)\left(a_{2}^{2} a_{3}+1\right) \cdots\left(a_{n}^{2} a_{1}+1\right)$$