Partition concentration conjecture for powers of paths

Fix integers $k\geq2$ and $m\geq k+1$. Let $P$ be an $m$-path whose vertex set is partitioned as

$$V(P)=V_1\cup\dots\cup V_{k+1}.$$

For a vertex subset $V_j$, write $P[V_j]$ for the induced subgraph, and let $f(\ell_{k,m})$ be the quantity defined in the paper. Partition concentration conjecture. There exists a positive constant $c=c(m)$ such that some $j\in[k+1]$ satisfies

$$|E(P[V_j])|\geq f(\ell_{k,m})|V_j|-c(m).$$

This is proposed as a strengthening of a lemma that would confirm the preceding over-threshold conjecture. Its status is open in the source.