Conjectural formulas for cluster variables of star graphs

Let $S_n$ be the star graph on $n$ vertices, with central vertex labeled by $1$, and let

$$\mathcal{C}=\{\{3\},\{4\},\dots,\{n\},\{1,3,4,\dots,n\}\}.$$

For a vertex subset $S$, write $Y_S$ for the associated cluster-algebra element and $[n]=\{1,\dots,n\}$. The cluster is $\mathcal{C}$, and the formulas below conjecturally describe $Y_S$ for every vertex subset $S$. Star-graph cluster formula conjecture. If $1\in S$, then

$$Y_S=\begin{cases} \displaystyle\frac{\prod\limits_{i\in S\backslash\{1,2\}}Y_i^2}{Y_{[n]\backslash\{1,2\}}}\left(\sum\limits_{i\notin S}\prod\limits_{j\ne1,2,i}Y_j\right)+\frac{Y_{[n]}}{\prod\limits_{i\in[n]\backslash S}Y_i}+\frac{Y_{[n]}}{Y_{[n]\backslash\{2\}}}\left(\sum\limits_{i\in[n]\backslash S}\frac{1}{Y_i}\right),&2\notin S,\\[2mm] \displaystyle\frac{Y_{[n]\backslash\{2\}}}{\prod\limits_{i\in[n\backslash(S\cup\{2\})]}Y_i}+\left(\prod\limits_{i\in S\backslash\{1\}}Y_i\right)\left(\sum\limits_{j\in[n]\backslash(S\cup\{2\})}\frac{1}{Y_i}\right),&2\in S. \end{cases}$$

If $1\notin S$, then $S$ consists either of disconnected leaves or of a single leaf, and

$$Y_S=\begin{cases}Y_i,&S=\{i\},\prod\limits_{i\in S}Y_i,&\text{otherwise.}\end{cases}$$

Here the source uses $\operatorname{textrm}{if}$ and contains an apparent index mismatch in the second case, with $Y_i$ occurring inside a sum indexed by $j$; these expressions are retained mathematically as stated. The conjecture proposes explicit formulas for cluster-algebra elements associated with subsets of a star graph and is presented as a future direction for extending rooted-cluster results to other clusters. No resolution is supplied in the source.