Fujimoto's polynomial general-position conjecture for even integers

Let $m\geq 2$ be an even integer, and set $t\coloneqq \frac{m}{2}$. Consider the following $3t$ polynomials:

$$\begin{aligned} &f_i(z)\coloneqq z^{i-1} &&(1\leq i\leq t),\\ &f_i(z)\coloneqq (z-1)^{i-1} &&(t+1\leq i\leq 2t),\\ &f_i(z)\coloneqq z^{i-t-1}(z-1)^{m-i+t} &&(2t+1\leq i\leq 3t). \end{aligned}$$

Fujimoto conjecture. These polynomials are in general position: any $m$ polynomials chosen from them are linearly independent over $\mathbb{C}$. This conjecture is the algebraic ingredient underlying constructions showing that the bound $\frac{m(m+1)}{2}$ for omitted hyperplanes is best possible in the even-dimensional cases; the paper proves it through total positivity, thereby completing the corresponding best-possible examples for all even integers $m\geq 4$.