  Calabi-Yau compactifications have typically a large number of complex
structure and/or K\"ahler moduli that have to be stabilised in
phenomenologically-relevant vacua. The former can in principle be done by
fluxes in type IIB solutions. However, the tadpole conjecture proposes that the
number of stabilised moduli can at most grow linearly with the tadpole charge
of the fluxes required for stabilisation. We scrutinise this conjecture in the
$2^6$ Gepner model: a non-geometric background mirror dual to a rigid
Calabi-Yau manifold, in the deep interior of moduli space. By constructing an
extensive set of supersymmetric Minkowski flux solutions, we spectacularly
confirm the linear growth, while achieving a slightly higher ratio of
stabilised moduli to flux charge than the conjectured upper bound. As a
byproduct, we obtain for the first time a set of solutions within the tadpole
bound where all complex structure moduli are massive. Since the $2^6$ model has
no K\"ahler moduli, these show that the massless Minkowski conjecture does not
hold beyond supergravity.
