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I am facing a n-dimensional fixed-point problem descending from a game theoretic problem given by the set of equations

$$ forall j in n: R_j (vecx_-j) - x_j = W left( A_j exp left(-sum_k neq j^n c_j,k x_kright)right) - x_j =0$$

where $W$ is the Lambert W function, $vecx$ is the vector of players' choices with $0 < x_j^- leq x_j leq x_j^+ < +infty $ $forall j$. The remaining parameters are given by $A_j > 0$ $forall j$ and $c_j,k in (0,1)$ $forall j$ $forall k neq j$.

I have shown concavity on the original problem and, given the bounds $vecx_j^-$, $vecx_j^+$, the Nash-Debreu-Theorem ensures existence of at least one fixed-point / equilibrium vector $vecx^*$.

What I am struggling with is showing uniqueness of the fixed-point of this set of equations.
I have tried using the contraction mapping (specifically Edelstein's Theorem) and univalent mapping approach (Gale & Nikaido (1965), Rosen (1965)) but could not find more than a sufficient condition for uniqueness so far.
Numerically uniqueness seems to hold. I have produced a large sample of these equilibrium problems and solved each from 1000 different starting vectors and each problem exhibited a single solution on $mathbbR_geq0^n$.

For the contraction mapping, since there is, to my knowledge, no identity for $W(z_1) - W(z_2)$, I had work with an upper bound on the integral representation of the difference which essentially gave me the diagonal dominance property of the $n times n$ matrix $mathbfC$ with elements $c_j,j=1$ and $c_j,k in (0,1)$ $forall j$ $forall k neq j$. Unfortunately I cannot make such a parametric restriction ex-ante which is why I believe the contraction mapping argument will not be helpful in proving this.

For the univalent mapping, Gale & Nikaido (1965) and Rosen (1965) guarantee uniqueness of the fixed-point if $mathbfH^* = frac12 (mathbfH+ mathbfH^T) $ is negative definite where $mathbfH$ is the $n times n$ Jacobian matrix of the n-dimensional root finding problem above. Note that the elements of $mathbfH$ are given by $h_j,j=-1$ and $h_j,k in (-1,0)$ since the derivative of the Lambert W $fracd Wd z leq 1$ on $z in mathbbR_geq0^$.

I would prove negative definiteness and thereby uniqueness of the fixed-point if I could show that

$$ vecz^T mathbfH^* vecz < 0 quad forall vecz in mathbbR_geq0^n setminus vec0 $$

which is equivalent to having all negative eigenvalues. However it is easy to find counterexamples with parameter combinations where $mathbfH^*$ has positive eigenvalues for $n > 2$.

My question is the following: The univalent mapping above, as I understand it, could give me uniqueness on the entirety of $mathbbR_^n$ but it does not hold generally in my case. My suspicion is the multiplicity of fixed-points depends on allowing negative values for $x_j$. Is there a property, let's call it constrained definiteness, I could apply to show univalence and therefore uniqueness just on $mathbbR_geq 0^n$ i.e. under the constraint $x_j geq 0$ $forall j$?