Signed enumeration of upper-right corners in path shuffles

We resolve a conjecture of Albert and Bousquet-Mélou enumerating quarter-planar walks with fixed horizontal and vertical projections according to their upper-right-corner count modulo 2. In doing this, we introduce a signed upper-right-corner count statistic. We find its distribution over planar walks with any choice of fixed horizontal and vertical projections. Additionally, we prove that the polynomial counting loops with a fixed horizontal and vertical projection according to the absolute value of their signed upper-right-corner count is (x+1)(x+1)-positive. Finally, we conjecture an equivalence between (x+1)(x+1)-positivity of the generating function for upper-right-corner count and signed upper-right-corner count, leading to a reformulation of a conjecture of Albert and Bousquet-Mélou on which their asymptotic analysis of permutations is sortable by two stacks in parallel relies.