I can't because twitch blocks the export when it mutes the audio, and I've never been able to get recording while streaming to work.
I could download the VOD in question using youtube-dl, but it does indeed have the silent parts as well. It's not just on playing it back on their website, sadly. That said, you could still make this stream available on YT as-is and make a note in the description that some parts of the audio are muted. Something's better than nothing, maybe?