We say $(f_1,f_2,\dotsc,f_N)$ is a convex conjugate if for any $i=1,2,\dotsc,N$ and any $x_i\in\Bbb R^d$, we have: $$f_i(x_i)=\sup\left\{\sum_{k=1}^{N}\sum_{j=k+1}^N x_k x_j - \sum_{j=1,j\neq i}^N f_j(x_j) \,\middle\vert\, \textrm{for all }j\neq i, x_j\in\Bbb R^d\right\}$$.
My question is for a given $n$-tuple $(g_1,g_2,\dotsc,g_N)$, if we check there is an index $i$ such that: $$g_i(x_i)=\sup\left\{\sum_{k=1}^{N}\sum_{j=k+1}^N x_k x_j - \sum_{j=1,j\neq i}^N g_j(x_j)\,\middle\vert\,\textrm{for all }j\neq i, x_j\in\Bbb R^d\right\}.$$
Can we say $(g_1,g_2,\dotsc,g_N)$ is a convex conjugate? That is, one condition implies the other conditions. If not, is there any counter-example?
The above is true when $N=2$, and was used as a known result for $N=3$ in Theorem 4.6 in the paper Multi-marginal maximal monotonicity and convex analysis by Bartz, Bauschke, Phan and Wang.
