The moderate doodle bordism extension conjecture for the annulus

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

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

The preceding discussion identifies the relevant invariant image with $\mathbb Z_2\times\mathbb Z$ and shows that the embedded subgroup is isomorphic to $\mathbb Z$; whether the resulting extension is bijectively described in this way is left as an open question.