Generalized spiral-point conjecture for Donaldson sections

For $k$ points on $S^2$, take generalized spiral points with cylindrical coordinates $(h_i,\theta_i)$, let $z_i$ be their complex coordinates, and define

$$p_k=e^{k/2}\prod_i \frac{(z-z_i)}{(1+|z_i|^2)^{1/2}},\qquad q_k=e^{k/2}\prod_i \frac{(z+z_i)}{(1+|z_i|^2)^{1/2}}.$$

Regard $(p_k\mathbf{w}^k,q_k\mathbf{w}^k)$ as sections and normalize them by a constant so that

$$\max\left(\lVert p_k\rVert^2+\lVert q_k\rVert^2\right)=1.$$

Generalized spiral-point conjecture. After this normalization, the sections $(p_k\mathbf{w}^k,q_k\mathbf{w}^k)$ satisfy the conditions in Theorem. The construction is supported by numerical verification and partial results, but the source gives no proof or resolution of the full assertion.