UGA VIGRE conjecture on the complexity of Specht modules

Let $S^\lambda$ be a Specht module in a block $B$ of weight $w$. Say that a partition is $p\times p$ if both it and its conjugate are of the form $p\tau$, equivalently, if its Young diagram is a union of $p\times p$ blocks. UGA VIGRE conjecture. The complexity of $S^\lambda$ is $w$ if and only if $\lambda$ is not $p\times p$. The complexity of a module is the least degree controlling the growth of the dimensions in a minimal projective resolution; the conjecture concerns when Specht modules attain the maximum complexity allowed by their block. It is motivated by the known maximal complexity of twisted Young modules and by limited results for simple modules, while the other direction remains open.