  We consider the limiting process that arises at the hard edge of
Muttalib--Borodin ensembles. This point process depends on $\theta > 0$ and has
a kernel built out of Wright's generalized Bessel functions. In a recent paper,
Claeys, Girotti and Stivigny have established first and second order
asymptotics for large gap probabilities in these ensembles. These asymptotics
take the form \begin{equation*} \mathbb{P}(\mbox{gap on } [0,s]) = C \exp
\left( -a s^{2\rho} + b s^{\rho} + c \ln s \right) (1 + o(1)) \qquad \mbox{as
}s \to + \infty, \end{equation*} where the constants $\rho$, $a$, and $b$ have
been derived explicitly via a differential identity in $s$ and the analysis of
a Riemann--Hilbert problem. Their method can be used to evaluate $c$ (with more
efforts), but does not allow for the evaluation of $C$. In this work, we obtain
expressions for the constants $c$ and $C$ by employing a differential identity
in $\theta$. When $\theta$ is rational, we find that $C$ can be expressed in
terms of Barnes' $G$-function. We also show that the asymptotic formula can be
extended to all orders in $s$.
