It can be shown (see Is every paracompact, Hausdorff, locally contractible space homotopy equivalent to a CW complex?) that if $X$ is a locally contractible paracompact Hausdorff space such that the $\infty$-topos $Sh(X)$ of sheaves on $X$ is hypercomplete then $X$ is homotopy equivalent to a CW complex. On the other hand, if $X$ is actually a CW complex then $X$ is a locally contractible paracompact Hausdorff space. However, it is not clear if in this case $Sh(X)$ is necessarily hypercomplete (it is true, for example, if $X$ is finite dimensional). Note that we are talking about the $\infty$-topos of all sheaves, not just the locally constant ones. The question then naturally arises - is $Sh(X)$ hypercomplete for every CW complex $X$?

It can be shown (see Is every paracompact, Hausdorff, locally contractible space homotopy equivalent to a CW complex?) that if $X$ is a locally contractible paracompact Hausdorff space such that the $\infty$-topos $Sh(X)$ of sheaves on $X$ is hypercomplete then $X$ is homotopy equivalent to a CW complex. On the other hand, if $X$ is actually a CW complex then $X$ is a locally contractible paracompact Hausdorff space. However, it is not clear if in this case $Sh(X)$ is necessarily hypercomplete (it is true, for example, if $X$ is finite dimensional). Note that we are talking about the $\infty$-topos of all sheaves, not just the locally constant ones. The question then naturally arises - is $Sh(X)$ hypercomplete for every CW complex $X$?