Extension of the finite-part regularity result to subcubic scales

Establish that the finite-part regularity assertion holds when the attached Young diagram has size \(k=o(n^{1/3})\), rather than only in the proven range \(k=o(n^{1/4})\).

Background

The paper studies diagrams formed by adjoining a smaller diagram of size $k$ to a large, regular diagram. Its proof establishes the result for $k=o(n^{1/4})$, while cited estimates indicate a counterexample at $k=n^{1/3}$ for a related skew-dimension asymptotic.

The conjecture seeks an improved range below the apparent cubic-root obstruction, asserting regularity throughout $k=o(n^{1/3})$.