The moderate open-curve doodle extension conjecture for the annulus

Let $A=\mathbb R\times[0,1]$. Denote by $\mathbf{OC}^{\mathsf{imm}}_{\mathsf{moderate \leq 2}}(A)$ the abelian group of moderate immersed open-curve doodle classes, and by $\mathbb Z_2$ the cyclic group of order $2$. The moderate open-curve doodle extension conjecture. There is a splittable abelian group extension

$$0\to\mathbb Z\to\mathbf{OC}^{\mathsf{imm}}_{\mathsf{moderate \leq 2}}(A)\to\mathbb Z_2\times\mathbb Z_2\times\mathbb Z\to0.$$

The proof establishes surjectivity of the invariant map onto $\mathbb Z_2\times\mathbb Z_2\times\mathbb Z$, while the displayed extension records the claimed splitting and the kernel identified with $\mathbb Z$.