It's been stated that the system requirement that says "Linux, 3.x and later" is conclusive of 4.x as well.
So I've basically confirmed that support will support you on kernel version 4.4.
Of course, one should always test said upgrade (freshening) before committing a production environment to the upgrade and/or version.
I will see what I can find out about 'support' - as for installing 6.5 on a Linux distribution with kernel 4.x, I've performed an installation of 64 bit Splunk 6.5.0 on Linux Mint 18 with a v4.4 kernel. No problems whatsoever thus far.