The paper suggests a resolution for recent controversy over convergence of heat conductivity in one-dimensional chains with asymmetric nearest-neighbor potential. We conjecture that the convergence is promoted not by the mere asymmetry of the potential, but due to ability of the chain to dissociate. In other terms, the attractive part of the potential function should approach a finite value as the distance between the neighbors grows. To clarify this point, we study the simplest model of this sort -- a chain of linearly elastic disks with finite diameter. If the distance between the disk centers exceeds their diameter, the disks cease to interact. Formation of gaps between the disks is the only possible mechanism for scattering of the oscillatory waves. Heat conduction in this system turns out to be convergent. Moreover, an asymptotic behavior of the heat conduction coefficient for the case of large densities and relatively low temperatures obeys simple Arrhenius-type law. In the limit of low densities, the heat conduction coefficient converges due to triple disk collisions. Numeric observations in both limits are grounded by analytic arguments. In a chain with Lennard-Jones nearest-neighbor potential the heat conductivity also saturates in a thermodynamic limit and the coefficient also scales according to the Arrhenius law for low temperatures. This finding points on a universal role played by the possibility of dissociation, as convergence of the heat conduction coefficient is considered.