Random Variable $X$ pairwise independent to Gaussian $Y, Z$ $\implies$ $X$ uncorrelated to $YZ$?

Question

for my Bachelor's Thesis I need a proof for the following lemma. I tried many ideas, but to no avail. Any ideas are greatly appreciated!

Let $X$ be a random variable and $Y,Z$ gaussian random variables.

$X$ pairwise independent to $Y, Z$ $\implies$ $X$ uncorrelated to $YZ$

I already know that $YZ = \frac{1}{4}(Y+Z)^2 - \frac{1}{4}(Y-Z)^2$ but I could not use that really. Basics in measure theory exist. Thank you for your help.

Answer 1

This is not true. You could have $X$ be the sign of $YZ$, where $Y$ and $Z$ have standard normal distribution.