Honestly, the other cat is too respectful of Jack. I have two cats, one respects the other, the other respects nothing: he would be all over Jack, using them as the pillow.
They definitely get catty over who gets other cat beds around the house but never the blue pillow which is Jack's all the way.
Could be several issues. Placement, scent, and individual preference all contribute to whether or not a cat likes a particular bed.
If you haven't already maybe try putting a worn tshirt on top of it so that it smells like you and put the bed in a place he normally likes to sit. If he still ignores it then there might just be something about that style or size of bed he doesn't like.
If he responds to catnip then sprinkling it on the bed might also get him to warm up to it.
I find it's a total crapshoot with my guys. Sometimes a comfy cardboard box becomes their new favorite place to sleep, sometimes the cat bed or macaron will be the primo spot they're both clamoring for.
It's a cat thing that our puny human brains can't comprehend.