The recursive formula for the Euclidean distance degree of plane circles

Let $C_m$ denote the variety associated with critical circles through a generic configuration of $m$ points in the plane, and let $EDD(C_m)$ denote its Euclidean distance degree. Define $a(n)$ by

$$a(n)=2^{n-3}(n^2+3n+4),$$

where $a(n)$ is the binomial transform of the modified triangular sequence. The circle EDD recurrence. The conjecture is

$$EDD(C_m)=EDD(C_{m-1})+a(m-1).$$

The formula is suggested by computational results for $m=3,\ldots,9$. Its validity beyond the displayed computations remains open.