Suppose that $X_1, X_2, Y$ are three scalar Gaussian distributed variables. Suppose we know that $X_1$ independent from $X_2$. However, $X_1, Y$ could be correlated. Similarly, $X_2, Y$ could be correlated. But we don't know $\rho_{x_1, y}$ and $\rho_{x_2, y}$.
My question is, the Pearson Correlation Coefficient between $(X_1, Y)$ and $(X_2, Y)$, which is $r_{x_1, y}, r_{x_2,y}$. Are these $2$ correlation coefficient independent?
I have run a ton of simulations and they all seem to be independent. But is there a way to prove their potential independence?