We say $(f_1,f_2,\cdots,f_N)$ a convex conjugate if for any $i=1,2,\cdots,N$ and any $x_i\in\Bbb R^d$, we have:
$$f_i(x_i)=\sup\left\{\sum_{k=1}^{N}\sum_{j=k+1}^N x_k x_j - \sum_{j=1,j\neq i}^N f_j(x_j)\, \mid\, \textrm{for all }j\neq i, x_j\in\Bbb R^d\right\}$$.

My question is for a given $n$-tuple $(g_1,g_2,\cdots,g_N)$, if we check there is an index $i$ such that:
$$g_i(x_i)=\sup\left\{\sum_{k=1}^{N}\sum_{j=k+1}^N x_k x_j - \sum_{j=1,j\neq i}^N g_j(x_j)\, \mid\,\textrm{for all }j\neq i, x_j\in\Bbb R^d\right\}$$.

Can we say $(g_1,g_2,\cdots,g_N)$ 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][1] by *Bartz, Bauschke, Phan and Wang*.

[1]: https://link.springer.com/article/10.1007/s10107-019-01433-9