system in $D$ spacetime dimensions, with action (in units where $8 \pi G = 1$), $$\begin{aligned}
S[g_{\alpha \beta}, \phi, A^{(j)}_{\gamma_1 \cdots \gamma_{n_j}}]
&=& S_E [g_{\alpha \beta}] + S_\phi[g_{\alpha \beta}, \phi]
+ \sum_{j=1}^k S_j[g_{\alpha \beta}, \phi, 
A^{(j)}_{\gamma_1 \cdots \gamma_{n_j}}]
+ \hbox{ ``more"},
\label{001}  \\
S_E [g_{\alpha \beta}] &=&{1 \over 2} \int R \sqrt{-g} \, d^D x,
\label{action2} \\
S_\phi[g_{\alpha \beta}, \phi]  &=& - {1 \over 2} \int 
\partial_\mu \phi \, \partial^\mu \phi  \sqrt{-g} \, d^D x, 
\label{action3} \\
S_j[g_{\alpha \beta}, \phi, A^{(j)}_{\gamma_1 \cdots \gamma_{n_j}}] &=&
- {1 \over 2 ( n_j + 1)!} \int F^{(j)}_{\mu_1 \cdots \mu_{n_j + 1}}
F^{(j) \, \mu_1 \cdots \mu_{n_j + 1}} e^{\lambda_j \phi} \sqrt{-g} \, d^D x,   
\label{action4}\end{aligned}$$ was investigated in the vicinity of a spacelike (“cosmological") singularity along the lines initiated by Belinskii, Khalatnikov and Lifshitz (BKL) [@BKL]. In (\[001\]), $g_{\alpha \beta}$ is the spacetime metric, $\phi$ is a massless scalar field known as the “dilaton", while the $A^{(j)}_{\gamma_1 \cdots \gamma_{n_j}}$ are a collection of $k$ exterior form gauge fields ($j=1, \cdots, k$), with exponential couplings to the dilaton, each coupling being characterized by an individual constant $\lambda_j$ (“dilaton coupling constant"). The $F^{(j)}$’s are the exterior derivatives $F^{(j)} = dA^{(j)}$, whereas “more" stands for possible 