Let $X$, $Y$ and $Z$ be independent, real-valued random variables, probably with continuous density functions. Define $A = X + Y$ and $B = X + Z$. Consider the regular conditional expectation $E_Y(a,b) = \mathbb E[Y|A=a, B=b]$.

Is it the case that $$\frac{\partial}{\partial a} E_Y(a,b) > 0 \quad \mathrm{and} \quad \frac{\partial}{\partial b} E_Y(a,b) \le 0?$$ If there is a counterexample, then are there simple conditions which guarantee this statement to be true?

**Edit:** fedja provides a counterexample in the case of bimodality. Assume that all the density functions in question are unimodal, or perhaps even log-concave. Under these assumptions, is it the case that the above partial differential inequalities hold?