The Extended Monotropic Programming problem deals with a more general extension to all n variables at a time than the two variables at a time extension you are interested in.
Specifically:
$\min_{x \in C} \Sigma_{i=1}^nf_i(x_i)$ with respect to $x$, where $x = (x_1,...,x_n)$, $x_i \in \mathbb{R}^{i}$
2010 postprint available at https://www.mit.edu/~dimitrib/Extended_Mono.pdf