  Tartaglia and Pearce have argued that the nonunitary $n\times n$ fused
Forrester-Baxter $\mbox{RSOS}(m,m')$ models are described, in the continuum
scaling limit, by the minimal models ${\cal M}(M,M',n)$ constructed as the
higher-level conformal cosets $(A^{(1)}_1)_k\otimes
(A^{(1)}_1)_n/(A^{(1)}_1)_{k+n}$ at integer fusion level $n\ge 1$ and
fractional level $k=nM/(M'\!-\!M)-2$ with $(M,M')=\big(nm-(n\!-\!1)m',m'\big)$.
These results rely on Yang-Baxter integrability and are valid in Regime III for
models determined by the crossing parameter $\lambda=(m'\!-\!m)\pi/m'$ in the
interval $0<\lambda<\pi/n$. Here we consider the $2\times 2$
$\mbox{RSOS}(m,m')$ models in the interval $\tfrac{\pi}{2}<\lambda<\pi$ and
investigate the associated one-dimensional sums. In this interval, we verify
that the one-dimensional sums produce new finitized Virasoro characters
$ch_{r,s}^{(N)}(q)$ of the minimal models ${\cal M}(m,m',1)$ with $m'>2m$. We
further conjecture finitized bosonic forms and check that these agree with the
ground state one-dimensional sums out to system sizes $N=12$. The $2\times 2$
$\mbox{RSOS}(m,m')$ models thus realize new Yang-Baxter integrable models in
the universality classes of the minimal models ${\cal M}(m,m',1)$. For the
series ${\cal M}(m,2m+1,1)$ with $m\ge 2$, the spin-1 one-dimensional sums were
previously analysed by Jacob and Mathieu without the underlying Yang-Baxter
structure. Finitized Kac characters $\chi_{r,s}^{m,m';(N)}(q)$ for the
logarithmic minimal models ${\cal LM}(p,p',1)$ are also obtained for $p'\ge 2p$
by taking the logarithmic limit $m,m'\to\infty$ with $m'/m\to p'/p+$.
