Fusion-limit conjecture for finitized principal subspaces

Let $A_{(m)}(n)$ be the subalgebra generated by $\xi_0,\ldots,\xi_{n-1}$ in $A_{(m)}$, and let $I_{(m)}(n)$ be its defining ideal. For a family $Z(\varepsilon)=(z_1(\varepsilon),\ldots,z_n(\varepsilon))\in\mathbb C^n$ satisfying the stated distinctness, convergence, and successive-spacing conditions, let $I_{(m),Z(\varepsilon)}(n)$ be the ideal generated by the corresponding pairwise products. Fusion-limit conjecture. If

$$\lim_{\varepsilon\to 0}\frac{z_2(\varepsilon)-z_1(\varepsilon)}{z_1(\varepsilon)}=0,\qquad \lim_{\varepsilon\to 0}\frac{z_{i+1}(\varepsilon)-z_i(\varepsilon)}{z_i(\varepsilon)-z_{i-1}(\varepsilon)}=0\quad (i=2,\ldots,n-1),$$

then

$$\lim_{\varepsilon\to 0}I_{(m),Z(\varepsilon)}(n)=I_{(m)}(n).$$

This generalizes the proved fusion-limit result for $m=1$ and would identify the fused finite-dimensional spaces with the desired finitizations for general $m$.