  We study compactifications of $6d$ non minimal $(D_{p+3},D_{p+3})$ type
conformal matter. These can be described by $N$ M5-branes probing a
$D_{p+3}$-type singularity. We derive $4d$ Lagrangians corresponding to
compactifications of such $6d$ SCFTs on three punctured spheres (trinions) with
two maximal punctures and one minimal puncture. The trinion models are
described by simple $\mathcal{N}=1$ quivers with $SU(2N)$ gauge nodes. We
derive the trinion Lagrangians using RG flows between the aforementioned $6d$
SCFTs with different values of $p$ and their relations to matching RG flows in
their compactifications to $4d$. The suggested trinions are shown to reduce to
known models in the minimal case of $N=1$. Additional checks are made to show
the new minimal punctures uphold the expected S-duality between models in which
we exchange two such punctures. We also show that closing the new minimal
puncture leads to expected flux tube models.
