What is Generalised forward market model?

The generalised forward market model (Lyashenko and Mercurio, 2019) models, for each accrual period $[T_{k-1},T_k]$, the forward $R_k(t) = \E^{T_k}_t[R(T_{k-1},T_k)]$ of the backward-looking rate, for all $t\le T_k$: before $T_{k-1}$ it coincides with the forward-looking forward of chapter 8, and during the period it keeps moving with the overnight fixings. Its volatility is $\sigma_k(t)g_k(t)$ with $g_k=1$ before $T_{k-1}$ and $g_k$ decaying to zero at $T_k$, for instance linearly.