In the absence of a genuine solution to the sign problem, lattice studies at imaginary quark chemical potential are an important tool to constrain the QCD phase diagram. We calculate the values of the tricritical quark masses in the Roberge-Weiss plane, $\mu=\imath\pi T/3$, which separate mass regions with chiral and deconfinement phase transitions from the intermediate region, for QCD with $N_\text{f}=2$ unimproved staggered quarks on $N_\tau=6$ lattices... A quantitative measure for the quality of finite size scaling plots is developed, which significantly reduces the subjective judgement required for fitting. We observe that larger aspect ratios are necessary to unambiguously determine the order of the transition than at $\mu=0$. Comparing with previous results from $N_\tau=4$ we find a $\sim50$% reduction in the light tricritical pion mass. The heavy tricritical pion mass stays roughly the same, but is too heavy to be resolved on $N_\tau=6$ lattices and thus equally afflicted with cut-off effects. Further comparison with other discretizations suggests that current cut-off effects on the light critical masses are likely to be larger than $\sim100$%, implying a drastic shrinking of the chiral first-order region to possibly zero. (read more)