Mean ergodic conjecture for locally semi-consistent extended spectral metric spaces

Let $(A,L)$ be a locally semi-consistent extended complete spectral metric space, and let $f:A\to M(A)$ be a relatively proper $*$-homomorphism. Write $T$ for the induced map and let $\mathcal{B}^{M(A)}$ denote the relevant Lipschitz ball. Suppose there exists $\Lambda\leq 1$ such that

$$L(T(b))\leq \Lambda L(b)$$

for all $b\in\mathcal{B}^{M(A)}$. Mean ergodic conjecture. For every $b\in\mathcal{B}^{M(A)}$, the sequence

$$\left(b_n:=\frac{1}{n+1}\sum_{i=0}^{n}T^i(b)\right)_{n=0}^{\infty}$$

converges strictly to some $\hat b$ satisfying $T(\hat b)=\hat b$. If $\Lambda<1$, then $L(\hat b)=0$. The conjecture is presented as a mean ergodic-style assertion for locally semi-consistent extended complete spectral metric spaces; its proof is described as mostly finalised and reserved for future work.