The polynomiality conjecture for double ramification cycles

Let ${\rm DR}_g(a_1,\dots,a_n)$ be the double ramification cycle in $\overline{\mathcal M}_{g,n}$ associated with labels $a_1,\dots,a_n$. The class has codimension $g$, so its Poincaré dual lies in $H^g(\overline{\mathcal M}_{g,n})$. Polynomiality conjecture. ${\rm DR}_g(a_1,\dots,a_n)$ is a polynomial in $a_1,\dots,a_n$ with coefficients in $H^g(\overline{\mathcal M}_{g,n})$. This folklore conjecture is motivated by Hain's polynomial formula on the compact-Jacobian locus and by the polynomiality observed in the intersection numbers studied in the paper; at the time of the source, the complete expression for the double ramification cycles remained unknown.