Eventual 4-periodicity conjecture for $C_p$-Mackey functors

Let $p$ be prime and let $M$ be a $C_p$-Mackey functor. A projective resolution of $M$ is eventually $4$-periodic if its differentials repeat with period $4$ from some point onward. Eventual 4-periodicity conjecture. Every $C_p$-Mackey functor admits an eventually $4$-periodic projective resolution. This is motivated by the $2$-periodicity of projective resolutions for $C_p$-modules and by experiments with 1000 randomly generated $C_p$-Mackey functors over various primes. The conjecture is presented as an experimental expectation, and no proof or disproof is given in the source.