Closed ideals in $\mathcal{L}(X)$ and $\mathcal{L}(X^{\ast })$ when $X$ contains certain copies of ${\mathrm{\ell }}_p$ and $c_0$

Suppose $X$ is a real or complexified Banach space containing a complemented copy of ${\mathrm{\ell }}_p$, $p\in (1,2)$, and a copy (not necessarily complemented) of either ${\mathrm{\ell }}_q$, $q\in (p,\mathrm{\infty })$, or $c_0$. Then $\mathcal{L}(X)$ and $\mathcal{L}(X^{\ast })$ each admit continuum many closed ideals. If in addition $q\ge p^{\mathrm{\prime }}$, $\frac{1}{p}+\frac{1}{p^{\mathrm{\prime }}}=1$, then the closed ideals of $\mathcal{L}(X)$ and $\mathcal{L}(X^{\ast })$ each fail to be linearly ordered. We obtain additional results in the special cases of $\mathcal{L}({\mathrm{\ell }}_1\oplus {\mathrm{\ell }}_q)$ and $\mathcal{L}({\mathrm{\ell }}_p\oplus c_0)$, $1<p<2<q<\mathrm{\infty }$.