Prove that if we directly homogenize the degree, it would be too high and difficult to handle. Consider using the conditions to gradually "increase the degree".
$$\begin{aligned} 9 \sum b^{4} c^{4}= & 9 \sum b^{3} c^{3} \cdot b c \leqslant 3 \sum b^{3} c^{3}\left(b^{3}+c^{3}+1\right)= \\ & 3 \sum b^{3} c^{3}\left(b^{3}+c^{3}\right)+3 \sum b^{3} c^{3}= \\ & \sum b^{3} c^{3}\left(b^{3}+c^{3}\right)+2 \sum b^{3} c^{3}\left(b^{3}+c^{3}\right)+3 \sum b^{3} c^{3} \leqslant \\ & \sum a^{9}+3 a^{3} b^{3} c^{3}+2 \sum b^{3} c^{3}\left(b^{3}+c^{3}\right)+ \\ & 3 \sum b^{3} c^{3}=(3 \text{rd degree Schur inequality}) \\ & 3\left(\sum a^{3}\right)^{2}=27 \end{aligned}$$

Thus, the proposition is proved!