Conditional convergence of $\Bbb E[X^{-1}]$ for $X\sim\mathcal{N}(\mu,\sigma^{2})$ as $\operatorname{pr}(X>0)\to 1$

This question references the post Reciprocal of a normal variable with non-zero mean and small variance

In the question the OP simulated observations from an inverse normal distribution, i.e. distribution of $X^{-1}$ where $X\sim\mathcal{N}(\mu,\sigma^{2})$, where $\mu$ and $\sigma$ were chosen such that $\operatorname{pr}(X>0)\approx1$. He noted that under these conditions the sample of inverse-normal pseudo random numbers he generated seem to be well behaved in the sense that he was able to compute a sample mean and variance that yielded reasonable values. That said, if $\mu$ is allowed to get small (or $\sigma$ is allowed to get large) such that $\operatorname{pr}(X>0)$ departs from $1$, the sample statistics become more and more ill behaved.

In the answer by to this post it was stated that this behavior was an example of conditional convergence. After checking out the wiki page on conditional convergence I am still unsure how this behavior is an example of it. I am looking for someone to (elaborate on)/(convince me) why mathematically this behavior is an example of conditional convergence. References supporting the answer would be nice too.