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, except that the literature I have seen deals only with the constraint set $C$ being a closed linear subspace, not a general convex set.
Specifically:
$\min_{x \in \text{closed linear subsapce}} \Sigma_{i=1}^nf_i(x_i)$ 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
Additional reference:
Older published version: https://link.springer.com/article/10.1007/s10957-010-9733-y