The closed-flow enumeration conjecture for forks

Let $\mathtt{Frk}_{i,n-i}$ be the fork with $n$ vertices and stem of size $i$, and let $\#\mathbb{F}(\mathtt{Frk}_{i,n-i},k,0)$ denote the number of closed flows of size $k$ on this fork. Closed-flow enumeration conjecture for forks.

$$\#\mathbb{F}(\mathtt{Frk}_{i,n-i},k,0)=\binom{i}{k}\binom{n}{k}-\binom{i+1}{k+1}\binom{n-1}{k-1}.$$

The formula proposes an explicit binomial enumeration of closed flows on forks. The source provides no resolution status.