  We explicitly determine the optimal degenerations of Fano threefolds $X$ in
family No 2.23 of Mori-Mukai's list as predicted by the Hamilton-Tian
conjecture. More precisely, we find a special degeneration $(\mathcal{X},
\xi_0)$ of $X$ such that $(\mathcal{X}_0, \xi_0)$ is weighted K-polystable,
which is equivalent to $(\mathcal{X}_0, \xi_0)$ admitting a K\"ahler-Ricci
soliton (KRS) by \cite{HL23} and \cite{BLXZ23}. Furthermore, we study the
moduli spaces of $(\mathcal{X}_0, \xi_0)$. The $\mathbf{H}$-invariant of $X$
divides the natural parameter space into two strata, which leads to different
moduli spaces of KRS Fano varieties. We show that one of them is isomorphic to
the GIT-moduli space of biconic curves $C\subseteq \mathbb{P}^1\times
\mathbb{P}^1$, and the other one is a single point.
