Uniqueness and iteration conjecture for the fixed point of $\hat{U}^{\dagger,0}$

Let $\hat{U}^{\dagger,0}:\mathbb{R}^N\to\mathbb{R}^N$ be the operator defined for the unknown stochastic shortest path problem, and let $\mathrm{CP}_{\mathrm{primalUnknown}}^{\dagger,\ell_1}$ denote the associated optimisation program. A fixed point of $\hat{U}^{\dagger,0}$ is a vector $x\in\mathbb{R}^N$ satisfying $\hat{U}^{\dagger,0}x=x$. Uniqueness and iteration conjecture. There exists a unique fixed point of $\hat{U}^{\dagger,0}$, which is equal to the unique optimal solution to $\mathrm{CP}_{\mathrm{primalUnknown}}^{\dagger,\ell_1}$. In almost all cases, this can be found by iterating the operator $\hat{U}^{\dagger,0}$ starting at any point in $\mathbb{R}^N$. This is presented as an open question; the source does not establish the claimed uniqueness or convergence, and notes that additional assumptions such as all policies being proper may be needed.