The adjusted generator bound for perfect ideals

Let $(R,\mathfrak{m})$ be a regular local ring, let $J$ be a perfect ideal of grade $g$, and let $n$ be a positive integer. Write $\mu(-)$ for the minimal number of generators. Assume that there is no ideal $I$ of height $g-2$ such that $J\subseteq I+\mathfrak{m}^n$.

The adjusted generator bound. If

$$\mu(J+\mathfrak{m}^n/\mathfrak{m}^n)\geq \begin{pmatrix} g+n-3\\ g-2 \end{pmatrix},$$

then

$$\mu(J)\leq \begin{pmatrix} g+n-2\\ g-1 \end{pmatrix}.$$

This is proposed as a modification of the earlier false conjecture; the source provides no proof or resolution.