Conjecture on the limiting genetic composition of an exponentially growing cell population

Let $\mathcal{N}$ be the set of nucleotides, let $\mathcal{S}$ be the set of genomic sites, and partition the neutral sites as

$$\mathcal{S}_\text{neut}=\bigcup_{j\in J}\mathcal{S}(j).$$

For $\chi,\psi\in\mathcal{N}$ with $\chi\not=\psi$, write $\mu^{\chi,\psi}(j)$ for the mutation rate at sites in $\mathcal{S}(j)$, and let $u(j)$ be the initial nucleotide at those sites. Let $\mathcal{T}$ be the tree indexing the random variables $P_x$, $R_x$, and $M_{x,r}$, and let $B_i^{n,\mu}$ denote the population count associated with site $i$ at population size $n$.

Limiting composition conjecture. Suppose

$$n\rightarrow\infty,\qquad n\mu^{\chi,\psi}(j)\rightarrow\theta^{\chi,\psi}(j)\in[0,\infty),\qquad \mu^{\chi,\psi}(j)|\mathcal{S}(j)|\rightarrow\eta^{\chi,\psi}(j)\in[0,\infty)$$

for all $j\in J$ and $\chi,\psi\in\mathcal{N}$ with $\chi\not=\psi$. Then

$$\sum_{i\in\mathcal{S}}\delta_{n^{-1}B_i^{n,\mu}}\rightarrow\sum_{x\in\mathcal{T}}\sum_{r=1}^{R_x}M_{x,r}\delta_{P_x}$$

in distribution, where convergence is in the same sense as in the preceding limit theorem. Here $(P_x)_{x\in\mathcal{T}\backslash\{\emptyset\}}$ has the distribution specified by the branching-process limit, with $P_\emptyset=1$; $(R_x)_{x\in\mathcal{T}\backslash\{\emptyset\}}$ is an i.i.d. family of geometric random variables with parameter $(\alpha(u')-\beta(u'))/(\alpha(u')+\beta(u'))$, while $R_\emptyset$ is independent of $(R_x)$ and satisfies $R_\emptyset\overset{d}{=}R_0-1$; $(P_x)$ is independent of $(R_x)$ if and only if $\beta(u')=0$; and $(M_{x,r})_{x\in\mathcal{T},r\in\mathbb{N}}$ is an i.i.d. family of Poisson random variables with mean

$$\sum_j\sum_{\psi\not=u(j)}\eta^{u(j),\psi}(j),$$

independent of $(P_x,R_x)$.