you'll need a good MHL adapter. they vary a lot so I would google around for one that has a proven reputation, irrespective of whether on Galaxy T4 or not.
I'm interested in the results. this has been an area strangely lacking scrutiny IMHO. There is a brilliant youtube/twitter/etc contributor going by the name of "Cold FusTion" that did some excellent demo work on the Note 3 when it first hit the streets, as it was a bit of a revolution tech-wise [4 not much better]. he demoes superb HDTV connectivity and way more, but aside from that one smart guy, the rule of thumb is "fail".
With PC's/laptops, this is so simple - even a bad laptop bought for $70 off ebay can usually sync right up - especially the AMD 'APU" based ones from the past 4 years or so.
in Android-land the "stick PC" has been around for years now but still very much a "toy" needing a lot of hands-on, custom ROMs, etc just to muster the mere basics