Schubert-flip Devron conjecture for configurations of lines

Let $L_1,\ldots,L_{2n}$, with $n\geq 6$, be a cyclically ordered sequence of lines in $\mathbb{R}^3$ such that each $L_i$ intersects $L_{i-3},L_{i-1},L_{i+1},L_{i+3}$. Assume that the even lines coincide:

$$L_2=L_4=\cdots=L_{2n}.$$

A Schubert flip replaces a line by the unique other line intersecting its four neighboring lines $L_{i\pm3}$ and $L_{i\pm1}$.

Schubert-flip Devron conjecture. Apply Schubert flips in a bipartite manner, beginning with the even lines. After $2n-7$ steps, one reaches another configuration in which half of the lines are equal.

This is the line-configuration analogue of the Devron property, with Schubert calculus supplying the two possible transversals to four generic lines in three-space. The source reports it as a system for which computer experiments suggest the property, so the assertion remains open.