  In recent work, we examined the algebraic structure underlying a class of
elements 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 Dunkl
operators, 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.
