The positive-half TQFT conjecture for $\mathfrak{psl}(1|1)$

Let $H=U(\mathfrak{psl}(1|1)^+)=\mathbb{C}[E]/(E^2)$, with coproduct and counit

$$\Delta(E)=E\otimes 1+1\otimes E,\qquad \varepsilon(E)=0.$$

The algebra $H$ has representation category $\operatorname{Rep}(H)$ with its tensor product.

Positive-half TQFT conjecture. The TQFT assigning $(\operatorname{Rep}(H),\otimes)$ to a point recovers both aspects of the $\mathfrak{psl}(1|1)$ Chern–Simons TQFT and aspects of decategorified bordered sutured Heegaard Floer theory in dimensions $1$ and $2$, thereby making these theories closely related.

This is motivated by the identification $U(\mathfrak{psl}(1|1))\cong D(U(\mathfrak{psl}(1|1)^+))$, which suggests that the obstruction to extending the corresponding Reshetikhin–Turaev theory to a point disappears in this case. The conjecture is explicitly described as imprecise, and the source gives no resolution.