Prove that using the conditions, equation (7) can be rewritten as
$$\begin{aligned} & \left(a-(a b c)^{\frac{1}{3}}+\frac{(a b c)^{\frac{2}{3}}}{b}\right)\left(b-(a b c)^{\frac{1}{3}}+\frac{(a b c)^{\frac{2}{3}}}{c}\right)\left(c-(a b c)^{\frac{1}{3}}\right. \\ + & \left.\frac{(a b c)^{\frac{2}{3}}}{a}\right) \leqslant a b c \end{aligned}$$

Then set $a = x^3$, $b = y^3$, $c = z^3$, where $x, y, z > 0$. Thus, equation (7) becomes
$$\begin{aligned} & \left(x^{3}-x y z+\frac{(x y z)^{2}}{y^{3}}\right)\left(y^{3}-x y z+\frac{(x y z)^{2}}{z^{3}}\right)\left(z^{3}-x y z\right. \\ + & \left.\frac{(x y z)^{2}}{x^{3}}\right) \leqslant x^{3} y^{3} z^{3} \end{aligned}$$

By Schur's inequality, we know

that is $\square$
$$3\left(x^{2} y\right)\left(y^{2} x\right)\left(z^{2} x\right)+\sum_{\text {cyc }}\left(x^{2} y\right)^{3} \geqslant \sum_{\text {sym }}\left(x^{2} y\right)^{2}\left(y^{2} z\right),$$
$$\begin{array}{l} \quad 3 x^{3} y^{3} z^{3}+\sum_{\text {cyc }} x^{6} y^{3} \geqslant \sum_{\text {cyc }} x^{4} y^{4} z+\sum_{\text {cyc }} x^{5} y^{2} z^{2}, \\ \left(x^{2} y-y^{2} z+z^{2} x\right)\left(y^{2} z-z^{2} x+x^{2} y\right)\left(z^{2} x-x^{2} y+y^{2} z\right) \\ \leqslant x^{3} y^{3} z^{3} . \end{array}$$

By equation (9), we know that equation (8) holds, thus equation (7) holds.