Given $X,Y,Z$ are independent random variables, are $X, Y$ conditional independent with $\mathbb{P}(\;\cdot\; |X\vee Y \leq Z)$?

Suppose $\mathbb{P} (X\vee Y \leq Z) > 0$, then by definition, $$\mathbb{P}(\{X\in B\} \cap \{Y\in C\} |X\vee Y \leq Z)\\ =\mathbb{P}(\{X\in B\} \cap \{Y\in C\} \cap \{X \leq Z\}\cap \{Y\leq Z\})/\mathbb{P} (X\vee Y \leq Z) $$ If $Z$ were a constant random variable, then this reduces to $$\mathbb{P}(\{X\in B\} |X\vee Y \leq Z) \cdot \mathbb{P}( \{Y\in C\} |X\vee Y \leq Z).$$ If it is true how could I show this?

A sufficient condition is $\mathbb{P} (X\leq Z, Y\leq Z) = \mathbb{P}(X\leq Z)\mathbb{P}(Y\leq Z)$, but this is false, take $X = Y = 1$.


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