The Z_p periodicity conjecture in the truncated case

Let $X^n$ be a space and let $p$ be a prime. Say that $H^*(X;\mathbb{Z}_p)$ is $k$-periodic up to degree $c$ when multiplication by a periodicity element induces successive isomorphisms through the last one landing in degree $c$. Assume that $k$ is the minimum period and that $0\leq m<k$ is the maximum degree with $H^m(X;\mathbb{Z}_2)\neq 0$. The truncated periodicity conjecture. If $p=2$ and $c\geq 2k+m$, then $k\in\{1,2,4\}$ or $(k,c)=(8,16+m)$. If $p\geq 3$ and $c\geq pk+m$, then $k=2\lambda$ for some divisor $\lambda$ of $p-1$. These are proposed truncated analogues of the periodicity conjectures; the source presents them as suggestions, and no resolution is given.