Do bonds with positive convexity fall slowly when interest rates rise?

Hello fellow monkeys,

I have a particular question about bond's convexity, most particularly for the FIG sector. I was reading this today:

"Positive convexity results in a higher present value of cash flows than duration alone would predict for an increase or decrease in rates. Negative convexity results is a lower present value of cash flows than duration alone would predict for an increase or decrease in rates."

It was accompanied by this graph:

My question is if interest rates decrease it will result in a higher value of cash flows than duration could predict but if it increases, would not (as per the graph) result in a higher value than duration could capture?

Hope you could shed some light on this