$\mu,\nu$ are fixed, which leads to a new particle system. Define the geometric-Bernoulli probability distribution[^5] on $k\in \mathbb{Z}_{\ge0}$ by $$\mathsf{gB}(\alpha,\beta;k)
    :=
    \begin{cases}
        \beta,& k=0;\\
        (1-\beta)(1-\alpha)\alpha^{k-1},& k\ge1.
    \end{cases}$$ We have for the two factors in the sum in : $$\label{q0_deg}
    \varphi_{q^{-1}, q^{g}, \mu \nu q^{g-1}}(p \mid \ell)
    \,\Big\vert_{q=0}
    =
    \begin{cases}
        1,&\textnormal{if $g=0$ and $p=\ell$};\\
        \dfrac{1}{1-\mu\nu},&
        \textnormal{if $0<g\le \ell$ and $p=\ell-g$};\\[9pt]
        \dfrac{-\mu\nu}{1-\mu\nu},&
        \textnormal{if $0<g\le\ell$ and $p=\ell-g+1$};\\[6pt]
        1,&\textnormal{if $g>\ell$ and $p=0$};\\
        0,&\textnormal{otherwise},
    \end{cases}$$ and $$\label{q0_deg1}
    \psi_{q, \nu \mu^{-1} q^{p}, \nu q^{g}, \nu^{2} q^{g+p}}(L-p)
    \,\Big\vert_{q=0}
    =
    \begin{cases}
        \mathsf{gB}(\mu,\frac{(1-\mu)(1+\nu)}{1-\mu\nu};L),
        & p=g=0;\\[6pt]
        \mathsf{gB}(\mu,\frac{1-\mu}{1-\nu};L),
        &p=0,\ g>0;\\[6pt]
        \mathsf{gB}(\mu,\frac{1-\mu}{1-\mu\nu};L-p),
        & p>0,\ g=0;\\[6pt]
        \mathsf{gB}(\mu,1-\mu;L-p),
        & p>0,\ g>0.
    \end{cases}$$ The first particle in this $q=0$ process jumps according to the distribution $\mathsf{gB}(\mu,\frac{1-\mu}{1-\nu};\cdot)$. To write down the update probabilities of all other particles, observe that the combination of the quantities – takes the form (recall that $\mathbf{P}_{\ell,g}(L)=0$ unless $L\ge \ell-g$) $$\label{q0_full_prob_P}
    \mathbf{P}_{\ell,g}(L)\,\big\vert_{q=0}=
    \begin{cases}
        \mathsf{gB}(\mu,\frac{1-\mu}{1-\nu};L)
        & \ell<g;
        \\[6pt]
        \mathsf{gB}(\mu,\frac{1-\mu}{1-\mu\nu};L-\ell+g)
        & \ell>g,\ L\ge \ell-g;
        \\[6pt]
        \mathsf{gB}(\mu,\frac{1-\mu}{(1-\nu)(1-\mu\nu)};L)
        & \ell=g>0;
        \\[6pt]
        \mathsf{gB}(\mu,\frac{(1-\mu)(1+\nu)}{1-\mu\nu};L)
        & \ell=g=0.
    \end{cases}$$ The condition that all the update probabilities $\mathbf{P}_{\ell,g}(L)\,\vert_{q=0}$ are nonnegative is equivalent to $0\le \mu<1$ and that all the second parameters of the geometric-Bernoulli distributions in are between $0$ and $1$. This leads to $$-1\le \nu\le 0, \ 0 \le \mu<1
    