Fair splitting into arbitrarily many stable sets for colored paths

Let $q$ be a positive integer, and let $P$ be a path whose vertex set is partitioned into $m$ subsets $V_1,\ldots,V_m$, each of size at least $q-1$. A subset of vertices is $q$-stable if no two of its vertices are at distance less than $q$, with distance counted in edges.

Fair stable-set splitting conjecture. There always exist pairwise disjoint $q$-stable sets $S_1,\ldots,S_q$ covering all vertices but $q-1$ in each $V_j$, with sizes differing by at most one, and satisfying

$$|S_i\cap V_j|\geq \left\lfloor \frac{|V_j|+1}{q}\right\rfloor-1$$

for all $i\in[q]$ and all $j\in[m]$.

This conjecture extends the paper's two-independent-set theorem to an arbitrary number of sets: when $q=2$, $q$-stable sets are precisely independent sets. Its resolution is not given in the supplied text.