  Let $R=k[x,y,z]$ be a standard graded $3$-variable polynomial ring, where $k$
denotes any field. We study grade $3$ homogeneous ideals $I \subseteq R$
defining compressed rings with socle $k(-s)^{\ell} \oplus k(-2s+1)$, where $s
\geq3$ and $\ell \geq 1$ are integers. The case for $\ell =1$ was studied in a
previous paper by the author; a generically minimal resolution was constructed
for all such ideals. More recently, this resolution is generalized in the guise
of (iterated) trimming complexes. In this paper, we show that all ideals of the
above form are resolved by an iterated trimming complex. Moreover, we apply
this machinery to construct ideals $I$ such that $R/I$ is a ring of Tor algebra
class $G (r)$ for some fixed $r \geq2$, and $R/I$ may be chosen to have
arbitrarily large type. In particular, this provides a new class of
counterexamples to a conjecture of Avramov not already constructed by
Christensen, Veliche, and Weyman.
