White's conjecture for lattice path polymatroids

Let $\Gamma$ be a lattice path polymatroid. Its base ring is the polynomial ring $\mathbb{C}[Y_m: m$ is a basis of $\Gamma]$, and its toric ideal is the kernel of the map

$$\phi: \mathbb{C}[Y_m: m \text{ is a basis of }\Gamma] \longrightarrow \mathbb{C}[x_0,x_1,x_2,\ldots],$$

defined by $\phi(Y_{m_1}Y_{m_2}\cdots Y_{m_t})=m_1m_2\cdots m_t$. A symmetric exchange binomial is a binomial $Y_mY_{m'}-Y_nY_{n'}$ obtained when $n$ and $n'$ are obtained from $m$ and $m'$ by a symmetric exchange. White's conjecture, adapted for polymatroids. The toric ideal of $\Gamma$ is generated by symmetric exchange binomials. The paper proves this statement for lattice path polymatroids, so it is solved in the stated class, although the broader polymatroid version is presented as an adaptation of White's conjecture.