Is the bounded rank perturbations problem for matrix pencils just a completion problem?

In this paper, we study a direct link between the bounded rank perturbations problem and the completion problem for matrix pencils. We conjecture that the bounded rank perturbations problem is, in fact, equivalent to a completion problem. We prove the conjecture in three cases: when the rank bound is one, when the involved pencils are of full row rank, and when the rank bound equals the rank distance of the involved matrix pencils.