  In 2006, Fock and Goncharov constructed a nice basis of the ring of regular
functions on the moduli space of framed ${\rm PGL}_2$-local systems on a
punctured surface $S$. The moduli space is birational to a cluster
$\mathcal{X}$-variety, whose positive real points recover the enhanced
Teichm\"uller space of $S$. Their basis is enumerated by integral laminations
on $S$, which are collections of closed curves in $S$ with integer weights.
Around ten years later, a quantized version of this basis, still enumerated by
integral laminations, was constructed by Allegretti and Kim. For each choice of
an ideal triangulation of $S$, each quantum basis element is a Laurent
polynomial in the exponential of quantum shear coordinates for edges of the
triangulation, with coefficients being Laurent polynomials in $q$ with integer
coefficients. We show that these coefficients are Laurent polynomials in $q$
with positive integer coefficients. Our result was expected in a positivity
conjecture for framed protected spin characters in physics and provides a
rigorous proof of it, and may also lead to other positivity results, as well as
categorification. A key step in our proof is to solve a purely topological and
combinatorial ordering problem about an ideal triangulation and a closed curve
on $S$. For this problem we introduce a certain graph on $S$, which is
interesting in its own right.
