The confuent hypergeometric function kind one distribution $CH(\nu,a,b)$ has density function at $x \geqslant 0$ given by $$ \frac{\Gamma(b-\nu)}{B(a-\nu,\nu)\Gamma(b)}x^{\nu-1} {}_1\!F_1(a, b; -x) $$ for $\underline{\nu > 0}$, $\underline{a > \nu}$, $\underline{b > \nu}$.
It is related to the Gamma-Beta distribution $\mathcal{G}\mathcal{B}(\nu, \alpha, \beta)$, which is the distribution of a random variable $X$ such that $$ (X \mid U=u) \sim \mathcal{G}(\nu, u), \quad U \sim \mathcal{B}(\alpha,\beta), $$ where the second parameter of the Gamma distribution $\mathcal{G}$ is its rate parameter, not the scale parameter.
Equivalently, the $\mathcal{G}{B}(\nu,\alpha,\beta)$ distribution is the distribution of the quotient $Y/U$ where $Y \sim \mathcal{G}(\nu,1)$ is independent of $U \sim \mathcal{B}(\alpha,\beta)$.
The relation between $CH$ and $\mathcal{G}{B}$ is $$ \boxed{CH(\nu, a, b) = \mathcal{G}{B}(\nu, a-\nu, b-a)} $$ under the parameter constraints $\nu>0$, $a>\nu$, $\underline{b > a}$, because $\mathcal{G}{B}(\nu, \alpha, \beta)$ is defined for $\beta>0$ only.
So we have an easy way to simulate $CH(\nu, a, b)$ when $\boxed{b>a}$. My question is: how to simulate this distribution under the general constraint $b > \nu$, allowing $b \leqslant a$?