Timing Belt Auto (Hydro) Tensioner

I have been fighting an issue with a rough idle at normal operating temperature, until the engine coolant temp reaches 165 F the engine idles slightly rough, once it hits normal operating temp, it begins to go into and out of a shudder. So pronounced that the steering wheel can be seen bouncing.

Stopped at a light with the AC on and in drive it is the worst, turn off AC and/or shift to neutral and it smooths out, increase RPMs and the shudder is gone.

This is an issue at idle only, accelerate and the shudder is not present.

I can go through a laundry list of items I have tested, replaced or eliminated as the cause.

My question is about the possibility that a weak Auto (Hydro) Timing Belt Tensioner is the cause of the rough idle when the engine reaches operating temp.

Background: The rough idle has been present for about 30K miles.

History:

At 70K the timing, balance belt and water pump were replaced at a local shop. Neither of the tensioners were replaced during the timing belt change.

Since it feels like a pulsation could this be a weak Auto-Tensioner pulley bouncing off of the belt?

Has anyone seen or heard of this? Any and all comments/suggestions will be appreciated