  A determinantal facet ideal (DFI) is an ideal $J_\Delta$ generated by maximal
minors of a generic matrix parametrized by an associated simplicial complex
$\Delta$. In this paper, we construct an explicit linear strand for the initial
ideal with respect to any diagonal term order $<$ of an arbitrary DFI. In
particular, we show that if $\Delta$ has no \emph{1-nonfaces}, then the Betti
numbers of the linear strand of $J_\Delta$ and its initial ideal coincide. We
apply this result to prove a conjecture of Ene, Herzog, and Hibi on Betti
numbers of closed binomial edge ideals in the case that the associated graph
has at most $2$ maximal cliques. More generally, we show that the linear strand
of the initial ideal (with respect to $<$) of \emph{any} DFI is supported on a
polyhedral cell complex obtained as an induced subcomplex of the \emph{complex
of boxes}, introduced by Nagel and Reiner.
