Here's my thinking:
Assume he's loose/passive calling station. If he's limped all 5 of the hands you've seen him in, I have to think that is a safe assumption.
I think of these types of players as:
* can't be pushed off draw
* not very aware of kickers
* Likes to get to SD because he'll assume we're bluffing
him a lot but probably only when he's hit some piece of the board.
If he hit a draw to get a big hand (is that only with A2 and 22) or if he hit trips, he'd bet. Wouldn't he? AJ would also bet if this line of reasoning makes sense.
If he hit a top pair/who cares what kicker, I could see him taking this line because he's passive.
If he hit nothing at all, would even a calling station keep calling with nothing? That makes me think that Kx and Qx are less likely.
So that leaves him with Ax and Jx.
More cards that we beat than cards we don't. I can't see that value betting thinly is wrong. If he raises (seems unlikely) then we fold.
Now let's have someone far wiser than me tell us if I'm even close in my thinking.