The extended conjecture on spheres attached to constrained immersed-curve spaces

Let $\mathbb{S}^2$ be the unit sphere, let $\rho_0$ be the prescribed curvature parameter, and let $\pmb{I}$ and $\pmb{Q}$ denote the endpoint data defining the space $\mathcal{L}^{+\kappa_0}_{-\kappa_0}(\pmb{I},\pmb{Q})$. For each $n\in\mathbb{N}$, define

$$D_{1,n}=d(p_1,q_2)+2n\pi,\qquad D_{2,n}=d(p_2,q_1)+2n\pi,$$ $$L_{1,n}=d(p_1,q_1)+2n\pi,\qquad L_{2,n}=d(p_2,q_2)+2n\pi.$$

Set

$$\bar{L}_{i,n}=2\left\lfloor\frac{L_{i,n}}{4\rho_0}\right\rfloor-1,\qquad \bar{D}_{i,n}=2\left\lfloor\frac{D_{i,n}}{4\rho_0}-\frac{1}{2}\right\rfloor.$$

Extended conjecture. If $\bar{L}_{i,n}>\bar{D}_{i,n}$ for $i=1,2$, then a sphere $\mathbb{S}^k$ is attached to $\mathcal{L}^{+\kappa_0}_{-\kappa_0}(\pmb{I},\pmb{Q})$, where $k=\bar{L}_{1,n}=\bar{L}_{2,n}$. If $\bar{D}_{i,n}>\bar{L}_{i,n}$ for $i=1,2$, then a sphere $\mathbb{S}^k$ is attached to the same space, where $k=\bar{D}_{1,n}=\bar{D}_{2,n}$. This extends the comparison between the truncated lengths associated with the endpoint pairs and predicts additional spherical attachments in the homotopy type of the constrained immersed-curve space; the source provides no resolution of the conjecture.