Conditional Independence Relations for $X_1\leftarrow X\rightarrow X_2$

Question

Let $X$ be a random variable, and let $X_1:=g_1(X)$ and $X_2:=g_2(X)$. Does it hold that $X\perp \!\!\! \perp X_1 | (X_1, X_2)$? (This statement is made in the proof of Proposition 1 in the appendix of Tschannen et al. https://openreview.net/pdf?id=rkxoh24FPH)

Answer 1

As already discussed in the comment, a constant random variable is independent of all random variables (see e.g. Independence between a constant random variable and another random variable.), and $X_1\mid(X_1,X_2)$ is constant.