  This paper develops our previous work on properness of a class of maps
related to the Jacobian conjecture. The paper has two main parts:
  - In part 1, we explore properties of the set of non-proper values $S_f$ (as
introduced by Z. Jelonek) of these maps. In particular, using a general
criterion for non-properness of these maps, we show that under a "generic
condition" (to be precise later) $S_f$ contains $0$ if it is non-empty. This
result is related to a conjecture in our previous paper. We obtain this by use
of a "dual set" to $S_f$, particularly designed for the special class of maps.
  - In part 2, we use the non-properness criteria obtained in our work to
construct a counter-example to the proposed proof in arXiv:2002.10249 of the
Jacobian conjecture.
  In the conclusion, we present some comments pertaining the Jacobian
conjecture and properness of polynomial maps in general.
