Conjecture Steady State from limit properties
Upgrading an exchange of comments, a critical point in the question is the expression
\begin{align}
\frac{\partial u(\tilde x)}{\partial \rho}=\frac{1}{(\rho + u'(\tilde x) + 1)^2} 
\end{align}

which is mistaken , because from 

\begin{align}
u(\tilde x) = \frac{\rho + u'(\tilde x)}{\rho + u'(\tilde x) + 1}.
\end{align}

we obtain

\begin{align}
\frac{\partial u(\tilde x)}{\partial \rho}=\frac{1+\partial u'(\tilde x)/\partial \rho}{(\rho + u'(\tilde x) + 1)^2} 
\end{align}

The OP notes that the sign of $\partial u'(\tilde x)/\partial \rho$ is indeterinate, and so $\partial u(\tilde x)/\partial \rho$ cannot be said to be monotonically increasing, which was what drove the pinning-down of the steady state.