Extremal ghost numbers for group algebras of p-groups

Let $p$ be a prime, let $k$ be a field of characteristic $p$, and let $G$ be a $p$-group of order $p^r$. Write $C_p$ for the cyclic group of order $p$, and $C_{p^r}$ for the cyclic group of order $p^r$. The notation $C_p^r$ denotes the elementary abelian $p$-group of rank $r$.

Ghost-number extremality conjecture.

$$\operatorname{ghost\ number} k(C_p^r) \leq \operatorname{ghost\ number} kG \leq \operatorname{ghost\ number} k(C_{p^r}).$$

This conjecture is the stabilized version of a preceding lemma. Together with the preceding comparison for groups of larger order, it would identify the elementary abelian group as having the smallest ghost number among groups of order $p^r$; in particular, for $p\geq 5$, it would imply that $l_3$ is the ghost number of $k(C_p^3)$, where $(l_1,l_2,l_3,\ldots)$ is the increasing sequence of ghost numbers of group algebras of $p$-groups. The supplied text gives no resolution, so the conjecture is recorded as open.