Why is this statement wrong? Convexity

Hi,

Can someone explain why the folowing statement is incorrect.

The effective convexity of a putable bond cannot be less than that of an otherwise identical option-free bond.

I always deal this question with the graph. If you can draw the graph of the curve for option free bond and putable bond, that should give the answer.

However, let me try to explain, first of all putable bonds will have always positive convexity.

Second, as the rates increase, the curve for putable bonds starts getting flatter, that implies it will have lower convexity at higher rates in comparison to the option free bond

thank you