Conway–Sloane fiber conjecture for dense sphere packings

Let $n$ satisfy $2\leq n\leq 8$, and let $2^k$ be the largest power of $2$ strictly less than $n$. A sphere packing is weakly recurrent and dense when it has the corresponding recurrence and density properties. Conway–Sloane fiber conjecture. Every weakly recurrent, dense sphere packing in dimension $2\leq n\leq 8$ fibers over one in dimension $2^k$. The conjecture is presented as a postulate inspired by Conway and Sloane, after a theorem showing that weakly recurrent dense packings have no gaps under suitable hypotheses; the supplied text gives no resolution of this stronger fiber statement.