We prove two recent conjectures of Bourn and Erickson (2023) regarding a certain family of polynomials N_n(x). The first conjecture says they have only real zeros, and the second concerns the sum of their coefficients. These polynomials arise as the numerators of generating functions in the context of the discrete one-dimensional earth mover's distance (EMD), and also have a connection to the Wiener index of minuscule lattices. Additionally, we prove that the coefficients of N_n(x) are asymptotically normal, the coefficient matrix of N_n(x) is totally positive, and the polynomial sequence N_n(x)_n≥1 is x-log-concave.