This is a follow up to the previous Question: Conditions for Linear Diophantine Equations to always have a solution
It was established in the above's answer that obtaining or testing for the existence of an solution of an ILP in $\mathbb N$ is an NP Hard Problem. This in turn makes obtaining solution for a (LDP) Linear Diophantine Problem (where $v$ > $n$) in $\mathbb N$ an NP-hard Problem too.
Now given the restricted ILP Problem with an additional constraint:
The number of unique pure constants (i.e. ones without variables, denoted by $c_0 below) permitted in any system has an fixed upper Bound (say 2) where equations are of the form:
$c_0 + c_1 x_1 + \dots + c_k x_k \ge 0$
Is this restricted version of ILP still NP-hard?