The naive generator bound for large truncations of perfect ideals

Let $(R,\mathfrak{m})$ be a regular local ring, let $J$ be a perfect ideal of grade $g>1$, and let $n$ be a positive integer. Write $\mu(-)$ for the minimal number of generators.

The naive generator bound. If

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

then

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

The paper explicitly states that this conjecture is false as written and later gives counterexamples, so the claim is refuted.