Jane Manchun Wong, reverse engineering expert in a report states that Instagram is testing an option that will allow users to share their Instagram live streams direct to IGTV as soon as the stream ends.
This would help build your IGTV content library. IGTV has not shown effective results since its implementation but Instagram in its recent announcements said that they will introduce monetization options for this platform which might give it a boost.
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IGTV has made significant consideration and if you can achieve that growth in a simple way and increase the value of your content then why will you not tap the switch and save your Instagram live streams. But then again the live streams are usually not the best productions. Some users might prefer downloading their live video and then edit it to improve the quality of the content before uploading it to IGTV. But not all users have time to edit the video or if you are happy with the originality of your video then this could be a great option.
There is no official date for the release of this update but the fact that Instagram is putting more effort into IGTV it is more likely that we will get this feature soon.