Let $K$ be a compact oriented 3-dimensional handlebody of genus $g$. The group $H_g$ of isotopy classes of diffeomorphisms of $K$ is called the handlebody group. (It embeds as a subgroup of the mapping class group of the genus $g$ surface $\partial K$.) The fundamental group of $K$ is a free group of rank $g$, so there is a homomorphism
$H_g \to Out(F_g).$
I've been thinking about this homomorphism and its kernel, and I've come to suspect that the kernel is generated by Dehn twists around curves in $\partial K$ that bound discs in $K$. These elements are all clearly contained in the kernel, but do they generate the entire kernel?
Does anyone know of a reference, proof, or counter example?