Connected four-reflection approximation conjecture for random reflectors

Let $M_n$ be sets of mirrors and let $\Lambda$ denote the measure on light rays. The sets $M_n$ are required to satisfy the conclusions of Theorem 16.2 except condition (b), namely the exceptional-set condition concerning rays that reflect exactly twice before returning to $L_*$. Connected four-reflection approximation conjecture. One can construct sets $M_n$ so that they satisfy Theorem 16.2 except for (b), are connected, and every light ray reflects from $M_n$ four times, except for a set of rays of $\Lambda$-measure $1/n$. This would strengthen the approximation theorem by imposing connectedness while prescribing four reflections for almost every ray; the source presents it as an expected consequence of iterating the construction, but gives no proof or resolution.