Property of zero mean and unit variance random variable
In general, a random variable with finite mean and variance does not even need to have a finite fourth moment. Even if we require a finite fourth moment your conjecture does not hold, as shown by this counterexample:
Let $Z$ follow a Student t-distribution with $\nu=5$. Then for $Y = \sqrt{\frac{3}{5}} Z$ we have $E[Y]=0$, $E[Y^2]=1$, and $E[Y^4]=9$, whereas $E[X^4]=3$ for $X \sim \mathcal{N}(0,1)$.
In other words, $Y$ is a distribution with positive excess kurtosis.