Fixed-size independent-set conjecture for highly connected chromatic graphs

Let $G$ be an $n$-vertex $k$-chromatic $\ell$-connected graph, and let $i_t(G)$ denote the number of independent sets of size $t$ in $G$. Fixed-size independent-set conjecture. If $3 \leq k \leq \ell$, $n \geq 2\ell$, and $t \geq 3$, then

$$i_t(G) \leq i_t(G^*).$$

This would extend the paper's result for sufficiently large independent sets to all sizes $t \geq 3$ under the stated parameter conditions; the conjecture is presented as an open question.