The cauliflower convergence conjecture for Euclid polynomial series

Let $E_k(\lambda)$ denote the Euclid polynomials, and consider the series

$$\frac{1}{\lambda}=\sum_{k=1}^{n}\frac{1}{E_k(\lambda)}+\frac{1}{E_{n+1}(\lambda)-1}.$$

The “cauliflower” is the region depicted in Figure. Cauliflower convergence conjecture. There is convergence outside the “cauliflower” and divergence inside the cauliflower. The preceding discussion establishes convergence for $\lambda>0$ and divergence for $\lambda=-\frac{1}{2}$, while the conjecture asserts the corresponding classification throughout the regions determined by the cauliflower.