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In recent work, we examined the algebraic structure underlying a class ofelements supercommuting with realization of the Lie superalgebra$\mathfrak{osp}(1|2)$ inside a generalization of the Weyl Clifford algebra.This generalization contained in particular the deformation by means of Dunkloperators, yielding a rational Cherednik algebra instead of the Weyl algebra.The aim of this work is to show that this is the full supercentralizer, give a(minimal) set of generators, and to describe the relation with the$(\mathrm{Pin}(d),\mathfrak{osp}(2m+1|2n))$ Howe dual pair.

Supercentralizers for deformations of the Pin osp dual pair