  In this work, we propose a $3D$ ensemble measure for center-vortex worldlines
and chains equipped with non-Abelian degrees of freedom. We derive an effective
field description for the center-element average where the vortices get
represented by $N$ flavors of effective Higgs fields transforming in the
fundamental representation. This field content is required to accommodate
fusion rules where $N$ vortices can be created out of the vacuum. The inclusion
of the chain sector, formed by center-vortex worldlines attached to pointlike
defects, leads to a discrete set of $Z(N)$ vacua. This type of SSB pattern
supports the formation of a stable domain wall between quarks, thus
accommodating not only a linear potential but also the L\"uscher term.
Moreover, after a detailed analysis of the associated field equations, the
asymptotic string tension turns out to scale with the quadratic Casimir of the
antisymmetric quark representation. These behaviors reproduce those derived
from Monte Carlo simulations in $SU(N)$ $3D$ Yang-Mills theory, which lacked
understanding in the framework of confinement as due to percolating magnetic
defects.
