  The thermal conductivity, $\kappa$, of a homogeneous chain of
generically-ranged interacting planar rotors, more precisely the inertial
$\alpha-XY$ model, is numerically studied with the coupling constant decaying
as $r^{-\alpha}$. The conductance ($\sigma\equiv \kappa/L$) is calculated for
typical values of $\alpha$, temperature $T$ and lattice size $L$. For large
$L$, the results can be collapsed into an universal $q$-stretched exponential
form, namely $L^{\delta_{\alpha}} \sigma \propto
e_{q_{\alpha}}^{-B_{\alpha}(L^{\gamma_{\alpha}}T)^{\eta_{\alpha}}}$, where
$e_q^z\equiv [1+ (1-q)z]^{1/(1-q)}$. The parameters
$(\gamma_{\alpha},\delta_{\alpha},B_{\alpha}, \eta_{\alpha})$ are
$\alpha$-dependent, and $q_{\alpha}$ is the index of the $q$-stretched
exponential. This form is achievable due to the ratio $\eta_{\alpha}/(q-1)$
being almost constant with respect to the lattice size $L$. Two distinct
regimes are identified: for $0 \le \alpha < 2$, the Fourier law is violated,
whereas for $\alpha\ge 2$, it is satisfied. These findings provide significant
insights into heat conduction mechanisms in systems with long-range
interactions.
