Fixed-size independent-set conjecture for chromatic graphs with connectivity below chromaticity

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 $k \geq 4$, $k>\ell$, and $t \geq 3$, then

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

This proposes that the same extremal behavior for the number of independent sets of size $t$ should persist when the chromatic number exceeds the connectivity; the source presents it as an open question.