Ranked cluster-variable counting formulas for Grassmannian cluster algebras

For positive integers $k,n,r$, let $N_{k,n,r}$ denote the number of cluster variables of rank $r$ in $\mathbb{C}[\operatorname{Gr}(k,n)]$. Cluster-variable counting conjecture. The corresponding numbers satisfy

$$N_{3,n,3}=24\binom{n}{8}+9\binom{n}{9},$$ $$N_{3,n,4}=288\binom{n}{9}+400\binom{n}{10}+264\binom{n}{11}+48\binom{n}{12},$$ $$N_{4,n,3}=174\binom{n}{8}+855\binom{n}{9}+1285\binom{n}{10}+693\binom{n}{11}+123\binom{n}{12}.$$

These formulas are extrapolated from computational results for Grassmannian cluster algebras and concern finite-rank slices of settings where the total number of cluster variables can be infinite; their general validity remains open.