Pullback injectivity conjecture for forgetful charts of prestable curves

Let $(g,n)\neq(1,0)$, and let $d\geq 0$ be fixed. For each $m\geq 0$, consider the forgetful morphism

$$F_m\colon \overline{\mathcal{M}}_{g,n+m}\to\mathfrak{M}_{g,n}.$$

It forgets the last $m$ markings without stabilizing the curve. Pullback injectivity conjecture. There exists $m_0\geq 0$ such that, for every $m\geq m_0$, the pullback

$$F_m^*\colon \mathrm{CH}^d(\mathfrak{M}_{g,n})\to\mathrm{CH}^d(\overline{\mathcal{M}}_{g,n+m})$$

is injective.

The images of the forgetful charts form an atlas of $\mathfrak{M}_{g,n}$, with complements whose codimension grows with $m$. The conjecture asserts that, in each fixed degree, sufficiently large stable-curve charts detect all Chow classes; the paper gives motivation and verifies cases in genus zero, but the general statement remains open.