  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.
