  The $T^{\prime}$ infinite layered nickelates have recently garnered
significant attention owing to the discovery of superconductivity in hole-doped
RNiO$_2$ (R $=$ La, Pr, or Nd), which is the $n = \infty$ member of the series
R$_{n+1}$Ni$_n$O$_{2n+2}$. Here, we investigate the $n = 3$ member, namely
R$_4$Ni$_3$O$_8$ (R $=$ La, Pr, or Nd) of this family. The compound
La$_4$Ni$_3$O$_8$ exhibits simultaneous charge/spin-stripe ordering at
T$_N^\ast$ $=$ 105 K, which is also concomitant with the onset of
metal-to-insulator (MIT) transition upon lowering the temperature below
T$_N^\ast$. We investigate the conspicuous absence of this transition in the Pr
and Nd analogues of La$_4$Ni$_3$O$_8$. To achieve this purpose, we synthesized
solid-solutions of the form (La, Pr)$_4$Ni$_3$O$_8$ and (La, Nd)$_4$Ni$_3$O$_8$
and examined the behavior of T$_N^\ast$ as a function of the average R-site
ionic radius ($r_{\overline{R}}$). We show that after an initial quasilinear
decrease with decreasing $r_{\overline{R}}$, T$_N^\ast$ suddenly vanishes in
the narrow range 1.134 $\AA$ $\leq$ $r_{\overline{R}}$ $\leq$ 1.143 $\AA$. In
the same range, we observed the emergence of a new transition below T$^\ast$,
whose onset temperature increases as $r_{\overline{R}}$ further decreases. We,
therefore, argue that the sudden vanishing of charge/spin-stripe/MIT ordering
upon decreasing $r_{\overline{R}}$ is due to the appearance of a new competing
phase. The point $r_{\overline{R}}$ $\approx$ $r_c$, where T$_N^\ast$ vanishes
and T$^\ast$ appears -- a quantum critical point -- should be investigated
further. In this regard, Pr$_4$Ni$_3$O$_8$ and Pr-rich samples should be useful
due to the weak magnetization response associated with the Pr-sublattice, as
shown here.
