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It has been conjectured that if 
$G= \mathop{({ \mathbb{Z} }_{p} )}\nolimits ^{r} $ acts freely on a finite 
$CW$-complex 
$X$ which is homotopy equivalent to a product of spheres 
${S}^{{n}_{1} } \times {S}^{{n}_{2} } \times \cdots \times {S}^{{n}_{k} } $, then 
$r\leq k$. We address this question with the relaxation that 
$X$ is finite-dimensional, and show that, to answer the question, it suffices to consider the case where the dimensions of the spheres are greater than or equal to 
$2$.
