On the isomorphism of the spaces $\mathcal{k}$ and $\mathcal{K}$

Let $1 \leqslant p \leqslant q < \infty$, let $\mathsf{X}$ be a separable $\mathcal{L}_p$-space and let $\mathsf{Y}$ be a separable $\mathcal{L}_q$-space. We show that the space $\mathfrak{L}(\mathsf{X},\mathsf{Y})$ of bounded linear operators from $\mathsf{X}$ into $\mathsf{Y}$ is isomorphic to $\m