Absence of irreducible multiple zeta-values in melon modular graph functions

The expansion of a modular graph function on a torus of modulus $\tau$ near the cusp is given by a Laurent polynomial in $y= \pi \Im (\tau)$ with coefficients that are rational multiples of single-valued multiple zeta-values, apart from the leading term whose coefficient is rational and exponentially suppressed terms. We prove that the coefficients of the non-leading terms in the Laurent polynomial of the modular graph function $D_N(\tau)$ associated with a melon graph is free of irreducible multiple zeta-values and can be written as a polynomial in odd zeta-values with rational coefficients for arbitrary $N \geq 0$. The proof proceeds by expressing a generating function for $D_N(\tau)$ in terms of an integral over the Virasoro-Shapiro closed-string tree amplitude.