Automorphic normalization of primitive idempotents

Determine whether, for a locally finite standard graded semiconnected algebra A and a complete set of primitive pairwise orthogonal idempotents {e_1, ..., e_m}, there exists an automorphism of A sending this set to the set of degree-zero components {(e_1)_0, ..., (e_m)_0}; also determine whether the conclusion remains valid when A is merely nonnegatively graded and locally finite.

Background

The main theorem shows that, under the paper's hypotheses, certain inhomogeneous primitive idempotents can be mapped to their degree-zero parts by an automorphism. The authors ask whether the idempotents must initially be homogeneous for some grading and whether the standardness assumption can be weakened to mere nonnegative grading.

References

Let $A_\bullet$ be a locally finite standard graded semiconnected algebra, and ${ e_1, \cdots, e_m }$ a complete set of primitive, pairwise orthogonal idempotents in $A$. Does there exist an automorphism that takes ${ e_1, \cdots, e_m }$ to ${ (e_1)0, \cdots, (e_m)_0 }$? If yes, is the same true for $A\bullet$ being nonnegatively graded and locally finite?

— Isomorphisms of graded semiconnected algebras  (2609.03288 - Dramburg, 3 Sep 2026) in Section 4, Further questions and observations, Question 4.2