The irreducible-component conjecture for the family $\nu_n(\alpha)$

Let $\mathbf{k}$ be the ground field, let $n\geq 1$, and let $\nu_n(\alpha)$ be the $n$-dimensional terminal algebra

$$\nu_n(\alpha)=\langle e,n_1,\ldots,n_{n-1}\rangle:\ e^2=e,\quad en_i=\alpha n_i,\quad n_i e=(1-\alpha)n_i\quad (i=1,\ldots,n-1;\ \alpha\in\mathbf{k}).$$

For an algebra $A$, write $O(A)$ for its orbit under the relevant change-of-basis group. Irreducible-component conjecture.

$$\overline{O\big(\nu_n(\alpha)\big)}$$

is an irreducible component of the variety of $n$-dimensional terminal algebras. This is the second conjecture proposed from the geometric classification of $2$-dimensional terminal algebras; its resolution is not given in the supplied text.