Does smooth convex functions with $C^1$ always have a strong convex conjugate function?

Does smooth convex functions with $C^1$ always have a strong convex
conjugate function?

Assume there is a convex function $F(x)$ whose first derivative is
continuous, does this mean $F(x)$ must have a strong convex conjugate
function $F^*$?
Is not, should $F(x)$ be more smooth which could enable a strong convex
conjugate function of itself?
I know thatC If $F$ is convex and has a Lipschitz continuous gradient
with modulus L, then $F^*$ is $1/L$-strongly convex.
Thank you!