Vertex disjoint simple paths of length k A lot of effort has been invested in finding simple k-paths, as well as in finding vertex disjoint paths.

Is there any known parametrized algorithm that given a graph $G=(V,E)$, decides whether there exist $p$ vertex disjoint simple paths of length $k$ in the graph?

Finding a single simple k-path is known to be single exponent $FPT$ for a long time, what about this variant (with respect to both $p,k$ as parameters)?
It appears that this problem (or a generalization of it) was considered with the Divide and Color approach yielding a $O^*(4^{(k-1)\cdot p})$ run-time algorithm for deciding the problem.