Positivity conjecture for completed 2-dimensional manifold pairings

Let $S$ be a compact $2$-dimensional surface, and let ${\mathcal M}_{S^2}^{\wedge}$ denote the $L^2$ completion of the vector space associated with the doubled surface $S^2$. Consider the quadratic function

$$|\langle {~}, {~} \rangle_{S}^{\wedge}|^2:{\mathcal M}_{S^2}^{\wedge}\rightarrow {\mathbb R}\cup\infty.$$

Positivity conjecture. For every compact $2$-dimensional surface $S$, this quadratic function has no kernel: if

$$|\langle {v}, {v} \rangle_S^{\wedge}|^2=0,$$

then $v=0$. The uncompleted positivity theorem is known in dimensions at most $3$, while positivity after completion is stated here as known for dimensions at most $2$; the issue is whether the corresponding positivity survives completion in dimension $3$.