  We present a conjecture for the massless sector of perturbative 4D $N=1$
heterotic $(0,2)$ string vacua, including $U(1)^n$ gauge symmetries,one of them
possibly anomalous (like in standard heterotic compactifications).
Mathematically it states that the positive hull generated by the charges of the
massless chiral multiplets spans a sublattice of the full charge lattice. We
have tested this conjecture in many heterotic $N=1$ compactifications in 4D.
Our motivation for this conjecture is that it allows to understand a very old
puzzle in $(0,2)$ $N=1$ heterotic compactification with an anomalous $U(1)$.
The conjecture guarantees that there is always a D-flat direction cancelling
the FI-term and restoring $N=1$ SUSY in a nearby vacuum. This is something that
has being verified in the past in a large number of cases, but whose origin has
remained obscure for decades. We argue that the existence of this lattice of
massless states guarantees the instability of heterotic non-BPS extremal
blackholes, as required by Weak Gravity Conjecture arguments. Thus the
pervasive existence of these nearby FI-cancelling vacua would be connected with
WGC arguments.
