ESI Widget stops working after several hours

I and others have indicated issues with ESI widgets:

https://wordpress.org/support/topic/widget-esi-not-working-disappear-when-enabled/
https://wordpress.org/support/topic/how-to-purge-esi-blocks-on-a-page-without-purging-whole-site-cache/

My current issue is I have two ESI widgets (public one shows new forum posts every 60 secs, other is a private one with ttl=0 that gets login user avatar) that work initially and for several hours, but then stop outputting. Enabling debugs and inspecting the page source indicates LS knows them, is trying to run them per TTL but not getting them to output.

My workarounds to this are currently:

1. go to the WP Widgets area, touch the widget parameters and save it. Then it works again.
2. purge the whole LS cache after which it too work again.
3. (haven’t tried it yet, but maybe restart LS cache plugin though likely lose cache in the process?)
4. (I haven’t tried yet, but assemble a number of such widgets, do an ob_start, capture buffer, check for content and round robin through until get one to output)

Is there some way I can automate coaxing the widget to work again without such extreme measures?

If I could automate whatever (1) does as a cron job say every 30 minutes to refresh these two problem ESI widgets that would be a adequate workaround so they could work without needing to be supervised every couple of hours.