Let $X$ and $Y$ be two random variables such that $X \sim \text{Gamma}(a, \lambda)$ and $Y \sim \text{Gamma}(b, \lambda)$. Let $W$ be a new random variable such that $W := \frac{X}{X + Y}$. If $X$ and $Y$ are independent then we can prove that $W \sim \text{Beta}(a, b)$. For example, we can compute the joint distribution of $W$ and $T := X + Y$ using the change of variables formula and then find the marginal distribution of $W$.
If $X$ and $Y$ are not independent however, what is the distribution of $W$? Is it still $\text{Beta}(a, b)$? If yes, how do I prove that $W \sim \text{Beta}(a, b)$? I tried conditioning on $Y$ and using the law of total probability, but I could not do much.