The conjectural description of the Grothendieck group of divided power algebras

Let $\bD$ be the divided power algebra over $\bk$, let $R$ be the relevant grading ring, and let $\cC_n$ be the category of finitely generated $\bk$-modules supported on $\bigcup_{n\mid h}V(\pi_h)$. Write $\cK_n=\rK_0(\cC_n)$, and let $\ul{\sK}$ be the quotient of

$$\bigoplus_{n\geq 1}\cK_n\otimes_{\bZ}R_n$$

by the relations

$$\frac{[M]_m}{1-t^n}=\frac{[M]_n}{1-t^n}$$

for $M\in\cC_m$ and $n\mid m$. Define the $R$-linear map $\varphi\colon\ul{\sK}\to\ul{\rK}(\bD)$ by

$$\varphi\left(\frac{[\bk/\fa]_n}{1-t^n}\right)=\frac{1-t^h}{1-t^n}[M(\fa,h)]$$

when $n\mid h$ and $\fa$ contains $\pi_h$. The conjectural description. The map $\varphi$ is an isomorphism. This would give a precise description of the Grothendieck group of finitely presented graded modules over the divided power algebra, refining the preceding expectation that the coefficients in Hilbert-series decompositions are well-defined in the groups $R\otimes_{\bZ}\rK_0(\cC_n)$.