Separation conjecture between semisaturation and packing numbers

Let $\mathrm{ssat}_r(K_{k+1})$ be the $r$-colour semisaturation parameter of $K_{k+1}$, and let $P_r(k)$ be the $r$-colour $k$-clique packing number. Let $r=r(k)\geq 3$ be any function of $k$, and write $f(k)\ll g(k)$ when $f(k)/g(k)\to 0$ as $k\to\infty$.

Separation conjecture. For any $r=r(k)\geq 3$,

$$\mathrm{ssat}_r(K_{k+1})\ll P_r(k)$$

as $k\to\infty$.

For fixed $k$ and $r\to\infty$, the paper proves an order-of-magnitude separation; this conjecture proposes the analogous separation in every remaining parameter range.