Is the handlebody group of **genus two** surface generated by Dehn twists along properly embedded disks and annuli?

Are there alternative ways to describe a set of generators that are conceptually simple (not necessarily finite but conceptually simple)? For this part, really I'm asking the question for the handlebody subgroup of the mapping class group so I am considering only the restriction of the homeomorphisms to the boundary of the handlebody. Therefore a description in terms of Dehn twists that are tractable is appreciated.

*Context*: Define a handlebody $B$ as a regular neighbourhood of a graph embedded in $\mathbb{R}^3$. The genus of a handlebody is defined as the rank of its fundamental group. The boundary of a genus $g$ handlebody is a surface of genus $g$ then. Let the handlebody group of genus $g$ be the group of homeomorphisms of the handlebody up to isotopy. For any properly embedded disk $D$ or annulus $A$ in $B$, one can define a positive Dehn twist along $D$ or $A$ in the standard way by either twisting once counter-clockwise around the center of a regular neighborhood of $D$ or twisting along the $S^1$ direction in a regular neighborhood of $A$.