A short variant of Pitt's theorem is the followig: for $1\leq p < r <\infty$ holds $$ \mathcal{B}(\ell_r(\mathbb{N}),\ell_p(\mathbb{N}))=\mathcal{K}(\ell_r(\mathbb{N}),\ell_p(\mathbb{N})) $$ Now let $S_1,S_2$ two infinite sets and at least one is uncountable. Is it still true that $$ \mathcal{B}(\ell_r(S_1),\ell_p(S_2))=\mathcal{K}(\ell_r(S_1),\ell_p(S_2)\quad ? $$

A short variant of Pitt's theorem is the followig: for $1\leq p < r <\infty$ holds $$ \mathcal{B}(\ell_r(\mathbb{N}),\ell_p(\mathbb{N}))=\mathcal{K}(\ell_r(\mathbb{N}),\ell_p(\mathbb{N})) $$ Now let $S_1,S_2$ two infinite sets and at least one is uncountable. Is it still true that $$ \mathcal{B}(\ell_r(S_1),\ell_p(S_2))=\mathcal{K}(\ell_r(S_1),\ell_p(S_2)\quad ? $$ I've tried to mimic standard proof of Pitt's from Albiac Kalton, but got stuck since unit ball of $\ell_r(S_1)$ with weak topology not necesseary metriazable.