Suppose I have two parametric integrals $F_i$ with $i=1,2$ defined as $$ F_i(z) := \int_\mathbb{R} dx f_i(x,z)$$ and suppose that $f_i$ are nice enough to ensure that the $F_i$ are holomorphic on some connected open subsets $U_i \subset \mathbb{C}$ (which could be different for each, and perhaps $U_1 \cap U_2=\emptyset$). Suppose further that each $F_i(z)$ admits a holomorphic continuation $\bar{F}_i(z)$ to some larger common domain $V$, which is also connected. Now consider the parametric integral $$ F_{1+2}(z) := \int_\mathbb{R} dx (f_1(x,z)+f_2(x,z))$$ and suppose there exists an open subset $U_{1+2} \subset \mathbb{C}$ on which it is holomorphic.
Question: Do we necessarily have $F_{1+2}(z) = \bar{F}_1(z) + \bar{F}_2(z)$ on $U_{1+2} \cap V$?
If yes, how to prove such a statement? If no, are there additional assumptions or tweaks that might make this statement true? Thanks in advance.
Edit: the statement as is seems to be wrong, judging by Example of an analytic continuation for a function in integral form. Can it still be saved somehow by strengthening assumptions? Maybe we can say that the maximal domain on which the parametric integral is holomorphic is connected?