The Geer–Young TQFT identification for punctured surfaces

Let $F_{g,p}$ be the surface of genus $g$ with $p\geq 1$ boundary components, viewed with all boundary components as punctures labeled by $V(0,0)_{\overline{0}}$. Let $\mathbf{Z}^{GY}$ be the Geer–Young TQFT, whose state space is graded by the second factor of $Z=\mathbb{Z}\times\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$. Let $\delta_{1/2}$ and $\pi_{1/2}$ be the degree-shift and parity functions defined above, and let $\mathbf{Z}^{S_+}_{\delta_{1/2},\pi_{1/2}}$ denote the corresponding shifted and parity-refined state space. Geer–Young TQFT identification. There is an isomorphism of $\mathbb{Z}$-graded super vector spaces

$$\mathbf{Z}^{S_+}_{\delta_{1/2},\pi_{1/2}}(F_{g,p})\cong\mathbf{Z}^{GY}(F_{g,p}).$$

This identifies the proposed surface-gluing construction with the Geer–Young theory for these punctured surfaces, including its grading and parity conventions. The source does not provide evidence that the asserted identification has been proved, so its status remains open.