Let $P_n(t)$ be polynomials with integer coefficients with $d_n = \deg(P_n(t))$ going to infinity when $n$ goes to infinity and with nonzero discriminants $disc(P_n(t)) \neq 0$.

Question: Is $$ \lbrace disc({P_n(t)})\rbrace ^{\frac{1}{d_n}} $$ bounded when $n$ goes to infinity ?