  In this paper, we propose a coupled system of complex Hessian equations which
generalizes the equation for constant scalar curvature K\"ahler (cscK) metrics.
We show this system can be realized variationally as the Euler-Lagrange
equation of a Hessian version of the Mabuchi K-energy in an infinite
dimensional space of $k$-Hessian potentials, which can be seen as an infinite
dimensional Riemannian manifold with negative sectional curvature. Finally, we
prove an a priori $C^0$-estimate for this system which depends on the Entropy,
which generalizes a fundamental result of Chen and Cheng for cscK metrics.
