Let $Q=[-1/2,1/2]^2$ be a unit square and let $(\ell_n,\varepsilon_n)_{n\geq1}$ be an iid sequence of isotropic lines intersecting $Q$ (more precisely, distributed according to a Haar measure on the space of lines) and colored indepently black ($\varepsilon_n=1$) or white ($\varepsilon_n=-1$). Color the vertical boundaries of $Q$ white and its horizontal boundaries black. The line $\ell_1$ dissects $Q$ into two polygons $Q_{1,1}$ and $Q_{1,2}$ whose boundaries are also colored black and white. Furthermore a polygon $Q_{1,i}$ is deleted, if the black points form a connected component of its boundary. Now iterate this procedure (i.e. take $\ell_2$, intersect it with all (maybe just one?) of the remaining polygons and delete some of the newly created polygons according to the above rule).

Does this procedure terminate a.s. after finitely many steps?

[this problem appeared while studying a Poisson line version of confetti-percolation, but it might also have some inherent appeal]