The Kerr–Griffiths–Green mixed Tate lifting conjecture for limit Hodge structures

Let $f:X\longrightarrow C$ be a proper semi-stable degeneration over a smooth curve $C/k$, with $k\subset\mathbb C$. Let $X_t$ be a fiber at a complex point $t\neq s$, and write its limit Hodge structure as

$$(H^n(X_t,\mathbb Q),W_\bullet,F_{\mathrm{lim}}^\bullet).$$

Assume that the monodromy

$$M:H^n_{Betti}(X_t,\mathbb Q)\longrightarrow H^n_{Betti}(X_t,\mathbb Q)$$

has only one Jordan block. Kerr–Griffiths–Green conjecture. The limit Hodge structure $(H^n(X_t,\mathbb Q),W_\bullet,F_{\mathrm{lim}}^\bullet)$ can be lifted to a mixed Tate motive over $k$. This concerns motivic lifts of Hodge-theoretic limits in degenerating families. The statement is attributed in the source to Kerr, Griffiths, and Green; no resolution is supplied in the excerpt.