The algebraicity conjecture for higher cycles in universal hypersurface families

Let $Y$ be the smooth projective variety and let ${\mathcal O}(1)$ be the chosen polarization. Let $T$ parametrize a family of hypersurfaces, let ${\mathcal X}_T$ be the corresponding universal hypersurface, and let ${\rm H}^{p+q}({\mathcal X}_T,\mathbb C)_v$ denote its vanishing cohomology with Hodge filtration $F^p$. For integers $p,q$, write $\overline{{\rm reg}}_{p,p-q}$ for the regulator map from higher Chow groups to vanishing cohomology.

The algebraicity conjecture. For all integers $b\in\mathbb N^*$ and $p\in\mathbb N$ there is an integer $D$ depending only on $b$, $p$, $Y$ and ${\mathcal O}(1)$, such that for all $d\geq D$ and $0\leq q\leq N-1$, if

$$c\leq b\frac{d^{N-q}}{(N-q)!}$$

then the map

$$\overline{{\rm reg}}_{p,p-q}:{\rm CH}^p({\mathcal X}_T,p-q)\otimes\mathbb C\to F^p{\rm H}^{p+q}({\mathcal X}_T,\mathbb C)_v$$

is surjective. By convention, ${\rm CH}^p({\mathcal X}_T,p-q)=0$ for $p<q$.

The conjecture predicts algebraicity of the relevant vanishing Hodge pieces just beyond the bounds arising from Nori-type connectivity results. It is known in the case $p+q=N$, $q\geq N/2$, provided the family lies in the locus parametrizing smooth hypersurfaces; the general case remains open.