Fixed-time marginal identity for the discrete-time particle system

Let $\boldsymbol{X}(n)$ be the particle configuration at time $n$, let $\boldsymbol{0}$ denote the densely packed initial configuration, and let $T^{\phi_{\alpha}^n}(\boldsymbol{0},\boldsymbol{\lambda})$ be the transition probability associated with the function $\phi_{\alpha}^n$. Fixed-time marginal identity. For any time $n\geq 0$,

$$\mathbb{P}(\boldsymbol{X}(n)=\boldsymbol{\lambda})=T^{\phi_{\alpha}^n}(\boldsymbol{0},\boldsymbol{\lambda}).$$

The identity is supported by numerical calculations for fixed-time marginals on multiple levels under densely packed initial conditions, but it is false without the fixed-time assumption.