Let $W$ and $S$ are two positive valued continuous random variable. Suppose the Spearman rank correlation between $W$ and $SW+g(S)$ is zero, where $g: [0,\infty)\rightarrow [0,\infty)$ is a convex function with a constraint that $g$ can't be of the form $g(x)=cx$, $c$ is a constant.
Is it possible, there exist some $\theta$ such that the following equations holds simultaneously,
\begin{equation} \begin{array}{rcl} &&P\left(S_1W_1+g(S_1)<S_3W_3+g(S_3),W_1<W_2<W_3\right)=\frac{1}{12},\; \mbox{and},\;\\[2ex] &&P\left(S_1W_1+\theta S_1<S_3W_3+\theta S_3),W_1<W_2<W_3\right)=\frac{1}{12}??. \end{array} \end{equation}
Here the first equation comes from the definition of population Spearman rank correlation for $(S_1,W_1),(S_2,W_2)$ and $(S_3,W_3)$ three i.i.d pairs of $(S,W)$.
My Try: Suppose there exist a $\theta$ for which the above holds, so we get $g(S)=\theta S$ (just comparing the above two equation), which is not admissible form for $g$. So can we claim here that there does not exist any $\theta$ for which the two equations holds.