Splitness of codimension- im(A_0) ideals

Determine whether every ideal J in a standard graded, locally finite, semiconnected algebra A_ullet satisfying dim(A/J) = dim(A_0) is split with respect to the grading.

Background

The proof distinguishes the positive-degree ideal A_+ from an arbitrary ideal J of the same codimension. A split ideal is one for which the inclusion A_0 22 A 22 A/J is an isomorphism, so that A_0 provides a complement to J. The authors establish this property for the ideal obtained by transporting the positive-degree ideal under an ungraded isomorphism, but do not establish it for all ideals of the same codimension.

References

However, it is not clear that every codimension \dim(A_0) ideal in $A$ is split.

Let $A_\bullet$ be standard graded, locally finite and semiconnected. Let $J \trianglelefteq$ be an ideal with $\dim(A/J) = \dim(A_0)$. Is $J$ necessarily split?

— Isomorphisms of graded semiconnected algebras  (2609.03288 - Dramburg, 3 Sep 2026) in Section 4, Further questions and observations, Question 4.1