We study the impact of a weak bond disorder on the spinon heat transport in the
S=1/2 antiferromagnetic (AFM) Heisenberg chain material ${\mathrm{Sr}}_{1\ensuremath{-}x}{\mathrm{Ca}}_{x}{\mathrm{CuO}}_{2}$. We observe a drastic suppression in the magnetic heat conductivity ${\ensuremath{\kappa}}_{\mathrm{mag}}$ even at tiny disorder levels (i.e., Ca-doping levels), in stark contrast to previous findings for ${\ensuremath{\kappa}}_{\mathrm{mag}}$ of
S=1/2 two-dimensional square lattice and two-leg spin-ladder systems, where a similar bond disorder has no effect on ${\ensuremath{\kappa}}_{\mathrm{mag}}$. Hence, our results underpin the exceptional role of integrability of the
S=1/2 AFM Heisenberg chain model and suggest that the bond disorder effectively destroys the ballistic nature of its heat transport. We further show that the suppression of ${\ensuremath{\kappa}}_{\mathrm{mag}}$ is captured by an effective spinon-impurity scattering length, which exhibits the same doping dependence as the long-distance exponential decay length of the spin-spin correlation as determined by density-matrix renormalization group calculations.
We consider a class of singlet resonanting-valence-bond wave functions on a square lattice, with the bond-length distribution as a variational parameter. This class contains the two limiting cases of the dimer wave function and of the N\'eel state. We present numerical calculations of the energy and the spin-spin correlation functions up to very large lattice sizes (180\ifmmode\times\else\texttimes\fi{}180) both for disordered states with exponentially decaying correlation functions and for ordered states. The energy of a disordered state can be within 0.1% of our best ordered state ($\frac{\ensuremath{-}0.3344J}{\mathrm{bond}}$).