A recent survey of a large sample of massive Galactic asymptotic giant branch (AGB) stars shows that significant overabundances of rubidium (up to 100 times solar), but merely solar zirconium, are present in these stars. These observations can set constraints on our theoretical notion of the slow neutron capture process (the s process) in AGB stars, as well as on the rates of the neutron capture reactions involved in the production of Rb and Zr. We use the Monash nucleosynthesis code with a recently extended network to try to reproduce these observations. We present results for AGB stars of masses 5, 6, and 6.5 Msolar and solar metallicity. We also show results for different available choices of the neutron capture rates, as well as for the possible inclusion of a partial mixing zone (PMZ), leading to the activation of the 13C neutron source. We find increasing Rb overabundances with increasing stellar mass, as observed, but we are far from matching the highest observed Rb enhancements. Inclusion of a PMZ increases the Rb abundance, but also produces an overabundance of Zr, contrary to what is observed. Only if the third dredge up efficiency remains as high as before the onset of the superwind phase during the final few pulses of a massive AGB star, can we match the highest [Rb/Fe] ratios observed by García-Hernández et al. [l]. A better understanding of the third dredge up efficiency with decreasing envelope mass for massive AGB stars is essential for further investigation of this issue.