I've not tried others, and there may be push back if I suggest this as a
solution, for various reasons.
I didn't suggest it as a solution; it's part of problem isolation.
And don't you have a development/test/staging environment which
replicates your production environment, where you can try things like
this? If not, creating one would be my very next act...
It may also be worth noting that it's https in production, and it
seems to keep connections alive for some clients but not others for
http, but closes for all on https.
Could be helpful -- with some detail.
What determines which requests go to https?
Which clients get the behavior you want, which don't, and how do the
request headers differ?