Convexity Q

I keep confusing myself with regards to convexity.

In times of interest rate vol- prefer convexity/barbell to the bullet. This is easy enough to memorize. But I want to confirm-
So if all rates are going down, we want the barbell exposure because of the long end exposure.
But if all rates are going up, we still prefer the barbell why? Don’t we want to avoid long end exposure?

You are correct on all points.

Your questions have more to do with duration than convexity. Remember that duration is a first-order effect while convexity is a second-order effect. In short, duration trumps convexity.