Suppose that we have two random variables defined on the same sample space $\Omega$

$X\sim \text{Hypoexp}(\alpha_1,\dots,\alpha_n)$ and $Y\sim \text{Hypoexp}(\beta_1,\dots,\beta_m)$

or, equivalently, that $X$ is distributed like a sum of $n$ exponentials with parameters $\alpha_1,\dots,\alpha_n$ and $Y$ like a sum of $m$ exponentials with parameters $\beta_1,\dots,\beta_m$. An explicit expression for the density of $X$ and $Y$ is well known (see e.g. "On the Convolution of Exponential Distributions" by M. Akkouchi).

If we also assume that $n>m$ and that for every realization $\omega\in\Omega$, $X(\omega)>Y(\omega)$, how can I derive the distribution of $X-Y$? In other words, how can I compute something like

$\mathbb{P}(X-Y< t\mid X>Y)$

knowing again that $n>m$?