For which $p$ and $r$ is the equation $\Pi_p(L^r,\cdot)=I_p(L^r,\cdot)$ true?

It is explained when the classes of $p$-absolutely summing and $p$-integral operators given on the space $L^r(\mu)$ coincide. For a Banach space $X$ there is considered the following subset of the real line:
$$ J_X\stackrel{\mathrm{def}}=\{p\colon1\le p<\infty,\ \Pi_p(X,Y)=I_p(X,Y)\ \forall Y\}. $$
In the case when $X$ is an infinite-dimensional subspace of the space $L^r(\mu)$, it is proved that $J_X=(1,2]$ if $1\le r\le2$, and $J_X=\{2\}$ if $2<r<\infty$ and $X$ is not isomorphic with a Hilbert space.