Extension conjecture for polarized variations of Hodge structure

Let $X$ be a smooth projective variety and let $D\subset X$ be an ample divisor. Let $\mathscr{H}\to D$ be a polarized variation of Hodge structure, and let its numerical invariants be fixed. Extension conjecture. If $\dim(D)$ is sufficiently large in terms of the numerical invariants of $\mathscr{H}$, then $\mathscr{H}$ extends uniquely to a polarized variation of Hodge structure on $X$. This generalizes the proved extension result for variations whose period maps target compact quotients of Hermitian symmetric domains or Shimura varieties of PEL type; the conjectural statement leaves the precise meaning of sufficiently large and the required generality to be established.