Grassmannian cluster algebra of infinite type has no trees in its mutation class

Question

The question is why the statement in the title is true (is it?).

To elaborate, recall that Grassmannian cluster algebra, according to Scott`s paper Grassmannians and Cluster Algebras, is the cluster structure on $\mathbb{C}[\operatorname{Gr}(k,n)]$, corresponding to the following quiver, called $\Gamma_{k,n}$:

In case $k=2$ this is a chain of length $n-3$, so the corresponding cluster algebra is of type $\mathsf{A}_{n-3}$. It is also not hard to show that for $k=3$ and $n=6,7,8$ a suitable sequence of mutations transforms $\Gamma_{k,n}$ into a tree, which happens to be a Dynkin diagram of type $\mathsf{D}_4$, $\mathsf{E}_6$ or $\mathsf{E}_8$, respectively. In all other cases this cluster algebras is of infinite type.

Now I ran some computer experiments and it seems that in the infinite case the mutation class of $\Gamma_{k,n}$ contains no trees at all.

Q1: Does this experimental observation hold in all cases? If so, why?

Q2: Can one at least find some nice representative in each mutation class? Say, having the smallest number of (unoriented) cycles.

Answer 1

Regarding question Q2, one can go a little bit further and describe simple diagrams with few edges for some more cases.

Let us talk about $Gr(p, p+q)$, so that there is a symmetry between $p$ and $q$. Assume moreover that $p\geq 2$ and $q\geq 2$.

For $p=2$ and $2\leq q$, one can mutate to the tree $A_{q-1}$.

For $p=3$ and $2\leq q\leq 7$, one can mutate to $A_2, D_4$, $E_6$, $E_8$, $E^{(1,1)}_8$ and $K_{12}$.

for $p=4$ and $2\leq q\leq 5$, one can mutate to $A_3, E_6$, $E^{(1,1)}_7$ and $W_{12}$.

The symbols $E^{(1,1)}_8$ and $E^{(1,1)}_7$ stand for quivers related to elliptic singularities. These quivers are obtained from a tree by replacing the unique triple point by a double edge surrounded by three oriented triangles.

The symbols $K_{12}$ and $W_{12}$ stands for quivers related to two of the 14 unimodal singularities of Arnold. Unoriented graphs describing these singularities can be found in many references about singularity theory. For a picture, see page 3 of arXiv:0708.0210. The quivers are almost trees, and only have one double edge and three oriented triangles.

EDIT (2022-08)

On may expect a similar relationship between the case $(p,q)=(3,11)$ and the exceptional bimodal singularity $E_{20}$ defined by $x^2+y^3+z^{11}$, for which a simple diagram is known by works of Ebeling. In this case, a sequence of mutations is yet to be found.