General conjecture for the stationary behavior of multi-floor TASEP

Consider a finite multi-floor TASEP with $c\ge 1$ floors and attempted particle arrival and departure rates $\alpha>0$ and $\beta>0$. Let

$$\zeta(\alpha,\beta,c)=\max_{1\le \ell\le c}\min\{\phi(\alpha,c-\ell+1),\phi(\beta,\ell)\},$$

and let

$$L^*=\operatorname*{\arg\,\max}_{1\le \ell\le c}\min\{\phi(\alpha,c-\ell+1),\phi(\beta,\ell)\}.$$

Assume condition (a): $L^*$ consists of a single element $\ell^*$ and $\phi(\alpha,c-\ell^*+1)\ne\phi(\beta,\ell^*)$; or condition (b): $\zeta(\alpha,\beta,c)=1/4$. General multi-floor TASEP conjecture. As $N\to\infty$, the limiting flux is $\zeta(\alpha,\beta,c)$. Under condition (a), assume without loss of generality that $\phi(\alpha,c-\ell^*+1)<\phi(\beta,\ell^*)$. There is one large stable zone on floor $\ell^*$, within which the stationary distribution converges to $\Psi_{\ell^*-1}\nu_\gamma$, where $\gamma<1/2$ satisfies

$$\gamma(1-\gamma)=\phi(\alpha,c-\ell^*+1)=\zeta(\alpha,\beta,c).$$

The left-side limit is ${\cal H}_l=\Psi_{\ell^*-1}{\cal L}(\alpha,c-\ell^*+1)$ and behaves like $\Psi_{\ell^*-1}\nu_\gamma$ at infinity. The right-side limit ${\cal H}_r$ has effective floor $c-\ell^*+1$, behaves like $[\Psi_{c-\ell^*}\nu_{1-\gamma}]^{\updownarrow}$ at infinity, and equals the limit for the one-sided system with $\ell^*$ floors, attempted hole arrival rate $\beta$, and attempted hole departure rate $\gamma$, lifted by $c-\ell^*$ floors. Under condition (b), there are exactly $|L^*|=c+2-c_\alpha-c_\beta$ large stable zones, of asymptotically equal size, on floors $\ell=c_\beta,\ldots,c-c_\alpha+1$, and within them the stationary distribution converges to $\Psi_{\ell-1}\nu_{1/2}$. The left-side asymptotic limit is ${\cal H}_l=\Psi_{c-c_\alpha}{\cal L}(\alpha,c_\alpha)$, and the right-side asymptotic limit, viewed from the perspective of holes, is

$${\cal H}_r=[\Psi_{c-c_\beta}{\cal L}(\beta,c_\beta)]^{\updownarrow}.$$

This conjecture predicts the limiting flux and the macroscopic stable-zone structure of the multi-floor TASEP, extending the corresponding one-sided behavior to finite systems; the parser supplies no evidence that it has been proved or disproved.