While the original paper of t-SNE is based on the SNE and SNE uses $\sigma_i^2$ (note the subscript $i$) while calculating the similarity of point $x_i$ WRT other points $j$. i.e., for each data point $x_i$, the corresponding $\sigma_i^2$ is used. But t-SNE uses only $\sigma^2$ (not $\sigma_i^2$) while calculating the similarity of every data point centred at $x_i$. This is even mentioned in the original paper of t-SNE (below Equation 3).
However, I found a lot of resources on t-SNE that claim $\sigma_i^2$ for data point centred at $x_i$ (not $\sigma^2$), which is completely contrary to what is used in the paper. However, I found this lecture mentioning the mathematics of t-SNE correctly (as used in the original paper).
What is actually going on?