  We give sharp conditions for the limiting Korn-Maxwell-Sobolev inequalities
\begin{align*}
  \lVert P\rVert_{{\dot{W}}{^{k-1,\frac{n}{n-1}}}(\mathbb{R}^n)}\le
c\big(\lVert\mathscr{A}[P]\rVert_{{\dot{W}}{^{k-1,\frac{n}{n-1}}}(\mathbb{R}^n)}+\lVert\mathbb{B}P\rVert_{L^{1}(\mathbb{R}^n)}\big)
  \end{align*} to hold for all $P\in C_{c}^{\infty}(\mathbb{R}^{n};V)$, where
$\mathscr{A}$ is a linear map between finite dimensional vector spaces and
$\mathbb{B}$ is a $k$-th order, linear and homogeneous constant-coefficient
differential operator. By the appearance of the $L^{1}$-norm of the
differential expression $\mathbb{B}P$ on the right-hand side, such inequalities
generalise previously known estimates to the borderline case $p=1$, and thereby
answer an open problem due to M\"{u}ller, Neff and the second author (Calc.
Var. PDE, 2021) in the affirmative.
