$\mathrm{E}\left[u u^{\mathrm{T}}\right]=\sigma^{2} I_{n} \text { is untrue } \iff \text{heteroskedasticity?}$ $\mathrm{E}\left[u u^{\mathrm{T}}\right]=\sigma^{2} I_{n} \text { is untrue } \iff \text{heteroskedasticity?}$
I know heteroskedasticity $\implies \mathrm{E}\left[u u^{\mathrm{T}}\right]=\sigma^{2} I_{n} \text { is untrue }$
I have no idea whether $\sigma^2I_n \neq \mathbb{E}[uu^T] \implies \text{heteroskedasticity}$
https://en.wikipedia.org/wiki/Heteroskedasticity-consistent_standard_errors

No. You could have serial correlation with homoskedasticity.
