Alishahi–Meunier conjecture on stable almost fair splittings of paths

Let $q$ be a positive integer, and let $P$ be a path whose vertex set is partitioned into $m$ subsets $V_1,\dots,V_m$, each of size at least $q-1$. A set of vertices is $q$-stable if any two distinct vertices in it are at distance at least $q$ in $P$. Alishahi–Meunier's conjecture. There exist pairwise disjoint $q$-stable sets $S_1,\dots,S_q$ covering all but $q-1$ vertices in each $V_j$, with sizes differing by at most one, and satisfying

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

for all $i\in\{1,2,\dots,q\}$ and $j\in\{1,2,\dots,m\}$. This conjecture asks for a particularly balanced stable splitting of a path; the source gives no resolution, so its status remains open.