Fix an integer $k\ge 1$. Let $X$ be a smooth manifold. It is well-known, that a real valued function on $X$ is smooth if its restriction along all smooth maps $M\to X$ for manifolds of dimension $\le k$ is smooth. A similar assertion seems to be true for $p$-forms on $X$ as long as $p<k$. My question concerns the case $p=k$.
Is it true that a $k$-form on $X$ is smooth if its restrictions along all smooth maps from manifolds of dimension $\le k$ are smooth.
The real question behind this is to understand the sheaf $i_*i^{*} \Omega$ on the category of smooth manifolds, where $i$ is the inclusion of the full subcategory of manifolds of dimension $\le k$ and $\Omega$ denotes the de Rham complex.