  According to the Grothendieck-Lefschetz theorem from SGA 2, there are no
nontrivial line bundles on the punctured spectrum $U_R$ of a local ring $R$
that is a complete intersection of dimension $\ge 4$. Dao conjectured a
generalization for vector bundles $\mathscr{V}$ of arbitrary rank on $U_R$:
such a $\mathscr{V}$ is free if and only if
$\mathrm{depth}_R(\mathrm{End}_R(\Gamma(U_R, \mathscr{V}))) \ge 4$. We use
deformation theoretic techniques to settle Dao's conjecture. We also present
examples showing that its assumptions are sharp and draw consequences for
splitting of vector bundles on complete intersections in projective space.
