Answer
Not conservative.
Work Step by Step
When $F(x,y)=ai+bj$ is a conservative field, then throughout the domain $D$, we get
$\dfrac{\partial a}{\partial y}=\dfrac{\partial b}{\partial x}$
Here, $a$ and $b$ are first-order partial derivatives on the domain $D$.
Then, we have $a_x=e^x \cos y; b_y=- e^x \cos y$
Here, $\dfrac{\partial a}{\partial y} \neq \dfrac{\partial b}{\partial x}$
Thus, the vector field $F$ is not conservative.