Suppose we have two random variables $X_1$ and $X_2$ that both have finite expectations, that may or may not be correlated, and that are known to be linearly related (i.e., I mean their sum forms a straight line of constant slope) according to the equation, $$ Y = aX_1 + bX_2 . $$ We know, by construction, that $E[Y|X_1=x_1]$ is is a straight line with constant slope. Then, when is $E[X_1|Y=y]$ also linear in the same sense?
Based on simulations, I believe that $E[X_1|Y=y]$ will be a straight line with constant slope if $X_1$ and $X_2$ are i.i.d., but I can't find a proof. It is certainly linear when $X_1$ and $X_2$ are both independent Normal.
One simple example of when $E[X_1|Y=y]$ is not linear (does not have constant slope) is when $X_1\sim Uniform[-c_1,c_1]$ and $X_2\sim Uniform[-c_2,c_2]$ when $c_1\neq c_2$. In this case, $E[X_1|Y=y]$ has a zigzag shape, increasing initially for small y then flattening out at zero for a while then increasing again.
So, is there a proof that $E[X_1|Y=y]$ is a straight line with constant slope for i.i.d. $X_1$ and $X_2$, and are there instances when $E[X_1|Y=y]$ is linear but $X_1$ and $X_2$ are not i.i.d.?
Any help is much appreciated!
EDIT: Here are examples below demonstrating a case when $E[X_1|Y=y]$ is not linear (i.e., does not form a straight line with constant slope despite $E[Y|X_1]$ being linear), specifically with $X_1 \sim Uniform[-2,2]$ and $X_2 \sim Uniform[-3,3]$ and coefficients a = b = 1. The first two plots are simulated data, where the black line is a moving average. The third plot is the actual numerical solution found by computing the conditional pdf and integrating.