Centralisers in Thompson's group $V_n$ In the article Centralizers in R. Thompson group $V_n$, the following question is asked:

Question: Is the centraliser of an element of $V_n$ always finitely presented?

I am wondering: is this question still open? I have a plausible argument in mind, but I would like to be sure that the answer is not already somewhere in the literature.
 A: I finally found the answer to my question. In Centralizers in R. Thompson group $V_n$, it is proved that the centraliser of an element of $V_n$ decomposes as 
$$ \left( \prod\limits_{i=1}^s K_{m_i} \rtimes G_{n,r_i} \right) \times \left( \prod\limits_{j=1}^t (( A_j \rtimes \mathbb{Z} ) \wr P_{j}) \right)$$
where 


*

*$A_1, \ldots, A_t, P_{1}, \ldots, P_t$ are finite groups;

*$K_{m_i} = \mathrm{Maps}(\mathfrak{C}_n, \mathbb{Z}_{m_i})^{r_i}$ where $\mathrm{Maps}(\mathfrak{C}_n,\mathbb{Z}_{m_i})$ is the group of continuous maps from the Cantor set $\mathfrak{C}_n$ to the cyclic group $\mathbb{Z}_{m_i}$ (endowed with the discrete topology) under pointwise multiplication;

*$G_{n,r_i}$ is the Higman-Thompson group.


The right factor is virtually $\mathbb{Z}^t$, so no problem here. And it follows from Theorem 4.9 in Cohomological finiteness conditions and centralisers in generalisations
of Thompson's group $V$ that the left factor is of type $F_\infty$ (i.e., it admits a classifying space containing finitely many cells in each dimension). 
It follows that centralisers in $V_n$ are not only finitely presented, they are of type $F_\infty$. 
