Reduction numbers for witnesses to the generalized Loewy length

Let $(R,\mathfrak{m})$ be a one-dimensional Cohen-Macaulay local ring. In this paper, we find the reduction number $r_z({\mathfrak{m}}^d)$ of ${\mathfrak{m}}^d$ with respect to a witness $z\in {\mathfrak{m}}^d\setminus {\mathfrak{m}}^{d+1}$ to the generalized Loewy length $\text{g}\mathrm{\ell }\mathrm{\ell }(R)$ for several infinite families of hypersurfaces $\{(R,\mathfrak{m})\}$. For every principal reduction $w$ of ${\mathfrak{m}}^d$, we have $\text{g}\mathrm{\ell }\mathrm{\ell }(R)\le d(r_w({\mathfrak{m}}^d)+1)$. We give examples of families $\{(R,\mathfrak{m})\}$ such that $d(r_z({\mathfrak{m}}^d)+1)-\text{g}\mathrm{\ell }\mathrm{\ell }(R)=0$ and $d(r_z({\mathfrak{m}}^d)+1)-\text{g}\mathrm{\ell }\mathrm{\ell }(R)=1$. Moreover, we show that the difference $d(r_z({\mathfrak{m}}^d)+1)-\text{g}\mathrm{\ell }\mathrm{\ell }(R)$ can vary independently of $\text{g}\mathrm{\ell }\mathrm{\ell }(R)-e(R)$.