  The symmetric and traceless part of the matrix element $\mathcal{S}
\mathcal{T} \langle N| \overline{q} D_{\mu} D_{\nu} q |N \rangle$ can be
determined from the second moment of the twist-3 parton distribution function
$e(x)$. Recently, novel experimental data on $e(x)$ have become available,
which enables us to evaluate the magnitude of the above matrix element with
considerably reduced systematic uncertainties. Based on the new experimental
data, we show that $\mathcal{S} \mathcal{T} \langle N| \overline{q} D_{\mu}
D_{\nu} q |N \rangle$ is likely to be at least an order of magnitude smaller
than what previous model-based estimates have so far suggested. We discuss the
consequences of this observation for the analysis of deep inelastic scattering
and QCD sum rules studies at finite density for the vector meson and the
nucleon, in which this matrix element is being used as an input parameter.
