  Let $R$ be a standard graded algebra over a field $k$ and $I$ be a
homogeneous ideal of $R$. We study the question whether there is a constant $c$
such that $\Soc(H^{j}_{\fm}(R/I^t))_{<-ct}=0$ for all $t\geq 1$ and a variation
of this question. We also draw a connection between this question and the
notion of gauge-boundedness in prime characteristic.
