  Algebraically fibering group is an algebraic generalization of the fibered
3-manifold group in higher dimensions. Let $M(\mathcal{P})$ and
$M(\mathcal{E})$ be the cusped and compact hyperbolic real moment-angled
manifolds associated to the hyperbolic right-angled 24-cell $\mathcal{P}$ and
the hyperbolic right-angled 120-cell $\mathcal{E}$, respectively.
Jankiewicz-Norin-Wise showed in [13] that $\pi_1(M(\mathcal{P}))$ and
$\pi_1(M(\mathcal{E}))$ are algebraic fibered. Namely, there are two exact
sequences $$1\rightarrow H_{\mathcal{P}}\rightarrow
\pi_1(M(\mathcal{P}))\xrightarrow{\phi_{\mathcal{P}}} \mathbb{Z}\rightarrow
1,$$ $$1\rightarrow H_{\mathcal{E}}\rightarrow
\pi_1(M(\mathcal{E}))\xrightarrow{\phi_{\mathcal{E}}} \mathbb{Z}\rightarrow
1,$$ where $H_{\mathcal{P}}$ and $H_{\mathcal{E}}$ are finitely generated. In
this paper, we furtherly show that the groups $H_{\mathcal{P}}$ and
$H_{\mathcal{E}}$ are not $FP_2$. In particular, those fiber-kernel groups are
finitely generated, but not finitely presented.
