The recursive formula for critical spheres over real configurations

Let $Z_m$ denote the variety of critical $(n-1)$-spheres associated with a generic configuration of $m$ points in $\mathbb R^n$, and let $EDdegree(Z_m)$ be its Euclidean distance degree. For $m>n+1$, the recursive formula for real critical spheres states

$$EDdegree(Z_m)=EDdegree(Z_{m-1})+\sum_{k=0}^{m}{m-1\choose k-1}\left(\frac{1+(-1)^k}{2}+\sum_{l=0}^{n}{k\choose l}-k-1\right)$$

and equivalently

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

The recurrence is presented as a conjecture after numerical computations for symmetric configurations. Whether it holds for all generic real configurations in the stated range remains open.