first prove by induction on $i$ that $s_i \leq |V_i|(r - i + 1)$ for $i = r, r-1, \ldots, 1$. In first place, if $i = r$, clearly $s_r = |V_r| = |V_r|(r - r + 1)$. If $i < r$, just two cases are possible. If $s_i = |V_i|$, clearly $s_i \leq |V_i|(r - i + 1)$. Otherwise, $s_i = 1 + s_{i+1}$. By the inductive hypothesis $s_{i+1} \leq |V_{i+1}|(r - i)$ and the fact that $|V_i| \geq |V_{i+1}|$, we obtain: $$s_i = 1 + s_{i+1} \leq 1 + |V_{i+1}|(r - i) \leq |V_{i+1}|(r - i + 1) \leq |V_i|(r - i + 1).$$ Hence, $\bigl\lceil \frac{s_i}{|V_i|} \bigr\rceil \leq r - i + 1 \leq r$ for all $i$ and therefore $\eta(G) \leq r$.

Since $\chi(G) \geq r$ for any complete $r$-partite graph $G$, we conclude that the conjecture holds for these graphs.

Join with complete graphs
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Let $G_1, G_2$ be disjoint graphs. The *join* of $G_1$ with $G_2$, denoted $G_1 \lor G_2$, is defined as the resulting graph $G'$ satisfying $V(G') = V(G_1) \cup V(G_2)$ and $E(G') = E(G_1) \cup E(G_2) \cup \{ (u,v) : u \in V(G_1),~ v \in V(G_2)\}$. Given a graph $G$, the 