Sandwiching the Lp norm sequence of random variable

Let X be a random variable with $\Vert X\Vert_p = E[\vert X\vert^p]^{1/p}<\infty$ for all $p\geq 1$.

Assume for all $1\leq q < r < s$,$$\lim_{p\to\infty} \frac{\Vert X\Vert_p}{p^{1/q}}=0$$and$$\limsup_{p\to\infty} \frac{\Vert X\Vert_p}{p^{1/s}}=\infty$$Does this imply that there is some $K>0$ such that$$\limsup_{p\to\infty} \frac{\Vert X\Vert_p}{p^{1/r}}=K$$