 Solution

No. ConsiderEvery 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 formpositive. 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.