= Solution
No. Consider
$$
(a_1,a_2,a_3,a_4)=(1,1,1,-4).
$$
Every finite coloring of the positive integers has an infinite color class. Choose $x<y$ in that class with $y/x$ as large as needed. Assigning $y$ to the three positive-coefficient variables and $x$ to the negative-coefficient variable makes the linear form
$$
3y-4x
$$
positive. Reversing the assignments makes it $3x-4y<0$. Thus both strict-sign hypotheses hold in every finite coloring.
The nonempty subset sums of $1,1,1,-4$ are among $1,2,3,-1,-2,-3,-4$, so none is zero. The <Rado theorem for one equation> therefore says that this coefficient vector is not partition regular.
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