Yip's lower-bound conjecture for the Tesler flow polytope

Let $\bm{N}=(1,1,\ldots,1,-n)$, and let $K_n(\bm{N})$ denote the number of lattice points of the corresponding Tesler flow polytope. Recall that $f_n\sim e^{1/2}\cdot n^{n-2}$ is the number of forests with vertex set $[n]$, equivalently the number of lattice points of the classical permutahedron.

Yip's conjecture. For $\bm{N}=(1,1,\ldots,1,-n)$, we have

$$K_n(\bm{N})\geq f_n.$$

Yip proposed this as a weaker statement than the conjecture that the Tesler polytope $\mathcal{F}_n(\bm{1})$ projects to the permutahedron. The bound would compare the lattice-point count of the Tesler flow polytope with the number of forests on $[n]$; the source provides no resolution of this conjecture.