Is the covariance of the squares always limited below by twice the covariance?

I found the following inequality in one of my calculations ($ X, Y $ are random variables centered):

$$ operatorname {E} (X ^ 2Y ^ 2) – operatorname {E} (X ^ 2) operatorname {E} (Y ^ 2) geq 2 operatorname {E} (XY) ^ 2 $$

or, written in terms of covariances,

$$ operatorname {Cov} (X ^ 2, Y ^ 2) geq 2 operatorname {Cov} (X, Y) $$.

Yes $ (X, Y) = (U, V) $ is a Gaussian centered in two dimensions, this becomes an equality and if $ (X, Y) = (H_p (U), H_q (V)) $, where $ (U, V) $ It remains a two-dimensional centered Gaussian with $ operatorname {E} (U ^ 2) = operatorname {E} (V ^ 2) = 1 $ Y $ H_k $ denotes the $ k $The polynomial of Hermite (probabilistic), the previous inequality is strict whenever $ p, q geq 2 $.

I have the feeling that something like this should be true for arbitrary random variables, but I could not prove it. Is this inequality familiar to someone or do you have an idea of ​​how you can prove / disprove?