Unclear inverse Z-transform of $G(z) = \frac{1-p}{z-p}$
As we've already seen in your previous question about that paper, the authors of that paper are not very much into details when it comes to math and signal processing. What Eq. $(10)$ shows is definitely not the inverse $\mathcal{Z}$ transform of

$$G(z)=\frac{1-p}{z-p}\tag{1}$$

which is

$$g[n]=(1-p)p^{(n-1)}u[n-1]\tag{2}$$

as you've figured out by yourself. What Eq. $(10)$ in the paper does show is the step response, given by

$$s[n]=\sum_{k=-\infty}^{n}g[k]=(1-p)\sum_{k=1}^np^{(k-1)}=1-p^n,\quad n\ge 0\tag{3}$$

So if you apply a step $\tilde{\mu}\cdot u[n]$ at the input, the system's response is

$$\mu[n]=\tilde{\mu}\left(1-p^n\right),\quad n\ge 0\tag{4}$$

as given by Eq. $(10)$ in the paper.