The isolation-number lower-bound conjectures

For a graph $G$, let ${\rm iso}(G,k)$ be the maximum number of pairwise disjoint $k$-clique isolating sets in a partition of $V(G)$, and let ${\rm iso}_c(G)$ be the analogous number for cycle isolating sets. The isolation-number lower-bound conjectures. (i) If $k \geq 3$ and $G$ is connected with $G \ncong K_k$, then

$${\rm iso}(G,k) \geq k+1.$$

(ii) If $G$ is connected with $G \ncong C_3$, then

$${\rm iso}_c(G) \geq 4.$$

These are explicitly stated to be equivalent to the two partition conjectures above, so they do not constitute additional independent claims.