Vanishing conjecture for holomorphic forms on \overline{\mathcal{M}}_{3,n}

Let $\overline{\mathcal{M}}_{3,15}$ and $\overline{\mathcal{M}}_{3,16}$ denote the moduli spaces of stable genus-$3$ curves with, respectively, $15$ and $16$ marked points. Vanishing conjecture. The spaces of holomorphic forms

$$H^{19,0}(\overline{\mathcal{M}}_{3,15})$$

and

$$H^{20,0}(\overline{\mathcal{M}}_{3,16})$$

vanish. This prediction would extend the description of the $\mathbf{FA}$-modules of holomorphic forms to degrees $19$ and $20$, conditional on additional vanishing statements in genus $3$; it is supported by further computations communicated privately by Bergström and Faber.