Is Var[max{ a, min{X, b}}] <= Var[X] for a < b and X being normally distributed?

Question

Given a normal random variable $X$, I define $Y \equiv$ max{ $a$, min{ $X, b$}} with $-\infty < a < b < \infty$. Then, is it true that $Var[Y] \leq Var[X]$? If so, please provide a proof. If not, please provide a counter-example.