Definition 1. A dendrite $X$ is a 1 dimensional retract of the closed unit disk. Equivalently, $X$ is a compact, locally path connected, and uniquely arcwise connected metric space.

Definition 2. The path $\alpha :[0,1]\rightarrow l_{2}$ “factors through a dendrite loop” if there exists a dendrite $X$, a loop $\gamma :[0,1]\rightarrow X$ (with $\gamma (0)=\gamma (1)$), and a map $\pi :X\rightarrow l_{2}$ such that $\alpha =\pi \gamma$.

Question. Suppose both the path $\alpha :[0,1]\rightarrow l_{2}$ and also the concatenated path $\alpha \ast \beta :[0,1]\rightarrow l_{2}$ can be factored through dendrite loops. Can $\beta $ be factored through a dendrite loop?

The following comments are intended to absorb some of the difficulties, and point to where the remaining difficulties lie.

The image of $\alpha$ can be an arbitrary Peano continuum embedded in Hilbert space $l_{2}$. However, all the difficult issues are present in the special case of the unit disk or unit cube.

Declare $\alpha :[0,1]\rightarrow l_{2}$ “trivial” if $\alpha $ factors through a dendrite loop. An elementary argument shows if the trivial loops $\alpha _{n} \rightarrow \alpha $ uniformly then $\alpha $ is trivial, i.e. the trivial loops in $l_{2}$ are closed in the uniform topology.

(Apply Ascoli's theorem and the fact that each trivial loop extends to a map of the unit disk, so that each component of each point preimage separates the disk into convex sets.)

Declare the path $\alpha $ to be “tree-trivial” if $\alpha $ lifts to a loop in a tree (a simply connected finite graph). An elementary proof shows $\beta $ is tree trivial if each of $\alpha $ and $\alpha \ast \beta $ is tree trivial.

An elementary proof shows if $\alpha $ is trivial then there exist tree-trivial loops $\alpha _{n}\rightarrow \alpha $.

(Lift $\alpha $ to a dendrite $X$ (a natural inverse limit of trees under strong deformation retraction, and project the retractions into $X$. We can even assume $X$ is an $R$-tree, i.e. a length space).)

Consequently the question at hand is equivalent to the following: Suppose $\alpha $ and $\alpha \ast \beta $ are trivial. Suppose $\alpha _{n}\rightarrow \alpha $ and $\alpha _{n}$ is tree trivial. Must there exist tree trivial loops $\alpha _{n} \ast \beta _{n}\rightarrow \alpha \ast \beta$?

Finally, it may be helpful consider the contrapositive, which can ultimately be reduced to the following:

Suppose the path $\alpha$ is irreducible, i.e. no subloop of the path $\alpha $ lifts to a dendrite loop. Suppose $\beta $ lifts to a dendrite loop. Can $\alpha \ast \beta $ lift to a dendrite loop?