If I have a diagonalizable matrix $A = V\Lambda V^{-1}$, is there a way to show that for any similar $B$ such that $B = T\Lambda T^{-1}$, the Euclidean condition number $\kappa_2(B) \geq \kappa_2(\Lambda)$? Or is this statement false?

(and if it's a well-known result, is there a good linear algebra reference that mentions this? I can't seem to find one.)