The recursive formula for critical spheres over complex configurations

Let $\mathcal H_{m,n}$ denote the variety associated with critical $(n-1)$-spheres for a generic configuration of $m$ points in $\mathbb C^n$. Here $EDdegree(\mathcal H_{m,n})$ denotes its Euclidean distance degree. For $m>n+2$, the recursive formula for critical spheres states that the number of critical spheres is

$$EDdegree(\mathcal H_{m,n})=EDdegree(\mathcal H_{m-1,n})+\sum_{k=0}^{m-1}{m-1\choose k}\sum_{l=0}^{n}{k+1\choose l}-2^{m-1}-m2^{m-2}.$$

The formula is proposed on the basis of numerical experiments and extends the explicitly proved case $n=1$, where the Euclidean distance degree is $2^{m-1}-1$. Its validity for general $n\geq 1$ remains open.