The nonvanishing solution conjecture for invertible complex matrices

Let $A=(a_{i,j})_{1\leq i,j\leq n}$ be a complex invertible matrix, and let $x_1,\dots,x_n$ be complex variables. Nonvanishing solution conjecture. The system

$$\left(\begin{matrix}x_1^2\\ x_2^2\\ \vdots\\ x_n^2\end{matrix}\right)=\left(\begin{matrix}a_{1,1}&a_{2,1}&\dots&a_{n,1}\\ a_{1,2}&a_{2,2}&\dots&a_{n,2}\\ \vdots&\vdots&\dots&\vdots\\ a_{1,n}&a_{2,n}&\dots&a_{n,n}\end{matrix}\right)\left(\begin{matrix}x_1\\ x_2\\ \vdots\\ x_n\end{matrix}\right)$$

has a solution $(x_1,\dots,x_n)$ with $x_i\neq0$ for every $i$. The conjecture is true for $n=1$ and $n=2$, as verified in the paper. It is introduced as the first of two related conjectures concerning subalgebras of evolution algebras; its general status is not resolved in the supplied text.