$Var(\sqrt{n}\left(\hat{\beta}_{F G L S}-\beta\right))=s^2_{FGLS}(\frac{1}{N} \mathbf{X}^{\prime} \hat{\Omega}^{-1} \mathbf{X})^{-1}$? If so, Wikipedia is wrong?
Wikipedia:

$\sqrt{n}\left(\hat{\beta}_{F G L S}-\beta\right) \stackrel{d}{\rightarrow} \mathcal{N}(0, \mathrm{p}-\lim \left(X^{\prime} \Omega^{-1} X / T\right))$

Powerll from University of California, Berkley:

$\sqrt{N}\left(\hat{\boldsymbol{\beta}}_{F G L S}-\boldsymbol{\beta}\right) \stackrel{d}{\rightarrow} N(\mathbf{0}, \operatorname{plim} s_{F G L S}^2\left(\frac{1}{N} \mathbf{X}^{\prime} \hat{\Omega}^{-1} \mathbf{X}\right)^{-1})$

Aside, I am not sure what $s^2_{FGLS}$ is.
https://eml.berkeley.edu/~powell/e240b_sp06/glsnotes.pdf
https://en.wikipedia.org/wiki/Generalized_least_squares
Convention equivalences
$T=N$
$\sqrt{N}=\sqrt{n}$

https://math.stackexchange.com/posts/593699/edit


$A = S D S^{-1}$,
then
$A^{-1} = (SDS^{-1})^{-1} = (S^{-1})^{-1} D^{-1} S^{-1} = S D^{-1} S^{-1}$.

At the outset, one of the confusions emanates from the two usage to state violation of homoscedasticity: Wikipedia follows $\operatorname{Cov}[\boldsymbol\varepsilon|\mathbf X] =\mathbf \Omega$ whereas many authors do prefer $\mathbb E\left[\boldsymbol\varepsilon\boldsymbol\varepsilon  ^\mathsf T |\mathbf X\right]=\sigma^2\mathbf\Omega.$
Few lines worth to spend:
If $\tilde{\boldsymbol\beta}$ is the GLS estimator of the parameters, then its covariance matrix is $\sigma^2\left(\mathbf X^\mathsf T\mathbf \Omega^{-1}\mathbf X\right) ^{-1}.$ Then its asymptotic covariance matrix is $$\frac{\sigma^2}T\lim_{T\to\infty}\left(\frac{\mathbf X^\mathsf T\mathbf \Omega^{-1}\mathbf X}T\right) ^{-1}.\tag 1$$ Also an unbiased, consistent, efficient as well as asymptotically efficient estimator of $\sigma^2$ is $$\tilde\sigma^2:=\frac{\tilde{\boldsymbol\varepsilon}^\mathsf T\mathbf \Omega^{-1}\tilde{\boldsymbol\varepsilon}}{T-K}.\tag 2$$ This is what Powell meant by $s^2_{\text{GLS}}.$ That is,
$$\operatorname{plim} s^2_{\text{GLS}} =\sigma^2.\tag{2.I}$$
Now, $$\tilde{\boldsymbol\beta}\sim\mathcal N\left(\boldsymbol\beta,\sigma^2\left(\mathbf X^\mathsf T\mathbf \Omega^{-1}\mathbf X\right) ^{-1}\right).\tag 3$$
Then,  as Powell showed,
$$ \sqrt T\left(\tilde{\boldsymbol\beta}-\boldsymbol\beta\right)\overset{\mathcal L}{\to}\mathcal N\left(\boldsymbol 0,\operatorname{plim}s^2_{\text{GLS}}\left(\frac{\mathbf X^\mathsf T\mathbf \Omega^{-1}\mathbf X}{T}\right) ^{-1}\right).\tag 4$$
In the same vein, when $\mathbf\Omega$ is not known, then its estimator $\hat{\mathbf \Omega}$ is used in place of $\mathbf \Omega$ and this is the essence of feasible GLS. Let the corresponding estimator be $\bar{\bar {\boldsymbol \beta}}.$
Now, \begin{align}\operatorname{plim}\sqrt T\left(\tilde{\boldsymbol\beta}-\boldsymbol\beta\right)&=\operatorname{plim}\left(\underbrace{\frac{\mathbf X^\mathsf T\mathbf \Omega^{-1}\mathbf X}T}_{\mathsf A2}\right) ^{-1}\operatorname{plim}\underbrace{\frac{\mathbf X^\mathsf T\mathbf \Omega^{-1}\boldsymbol\varepsilon}{\sqrt T}}_{\mathsf B2},\tag{5.I}\\\operatorname{plim}\sqrt T\left(\bar{\bar{\boldsymbol\beta}}-\boldsymbol\beta\right)&=\operatorname{plim}\left(\underbrace{\frac{\mathbf X^\mathsf T\hat{\mathbf \Omega}^{-1}\mathbf X}T}_{\mathsf A1}\right) ^{-1}\operatorname{plim}\underbrace{\frac{\mathbf X^\mathsf T\hat{\mathbf \Omega}^{-1}\boldsymbol\varepsilon}{\sqrt T}}_{\mathsf B1};\tag{5.II}\end{align}
$\operatorname{plim}\mathsf{A2}= \operatorname{plim}\mathsf{A1}$ and $\operatorname{plim}\mathsf{B2}= \operatorname{plim}\mathsf{B1}$ ensure that (in fact, sufficient) FGLS estimator and GLS estimator have the same asymptotic distribution whence $\sqrt T\left(\tilde{\boldsymbol\beta}-\boldsymbol\beta\right)$ and $\sqrt T\left(\bar{\bar{\boldsymbol\beta}}-\boldsymbol\beta\right)$ have the same asymptotic distribution $\mathcal N\left(\boldsymbol 0,\sigma^2\mathbf Q^{-1}\right)$ where $\mathbf Q:= \operatorname{plim}\left(\frac{\mathbf X^\mathsf T\mathbf \Omega^{-1}\mathbf X}{T}\right) . $
So, Powell is correct. He used $\operatorname{plim} s^2_{\text{FGLM}} =\sigma^2.$

References:
$[1]$ Econometrics, Peter Schmidt, Taylor & Francis Group, $1976.$
$[2]$ Econometrics, Badi H. Baltagi, Springer-Verlag, $2008.$
