Must a finitely presented nil algebra be nilpotent? (Ufnarovskij's question)

Claims Corollary 3.8: there is an infinite-dimensional finitely presented nonunital associative $\mathbb F_2$-algebra that is nil and Jacobson radical but not nilpotent; its unitization is finitely presented, algebraic and infinite-dimensional. This answers Ufnarovskij's nil question, Amitsur's radical question and the finite-presentation version of Kurosh's algebraic finiteness question negatively, over $\mathbb F_2$. Every matrix over its positive-degree ideal is nilpotent. It does NOT give examples over other fields or in characteristic zero, and nilpotence indices are unbounded.

Claims Corollary 3.8: there is an infinite-dimensional finitely presented nonunital associative $\mathbb F_2$-algebra that is nil and Jacobson radical but not nilpotent; its unitization is finitely presented, algebraic and infinite-dimensional. This answers Ufnarovskij's nil question, Amitsur's radical question and the finite-presentation version of Kurosh's algebraic finiteness question negatively, over $\mathbb F_2$. Every matrix over its positive-degree ideal is nilpotent. It does NOT give examples over other fields or in characteristic zero, and nilpotence indices are unbounded.