  We establish new results on the $\mathcal I$-approximation property for the
Banach operator ideal $\mathcal I=\mathcal{K}_{up}$ of the unconditionally
$p$-compact operators in the case of $1\le p<2$. As a consequence of our
results, we provide a negative answer for the case $p=1$ of a problem posed by
J.M. Kim (2017). Namely, the $\mathcal K_{u1}$-approximation property implies
neither the $\mathcal{SK}_1$-approximation property nor the (classical)
approximation property; and the $\mathcal{SK}_1$-approximation property implies
neither the $\mathcal{K}_{u1}$-approximation property nor the approximation
property. Here $\mathcal{SK}_p$ denotes the $p$-compact operators of Sinha and
Karn for $p\ge 1$. We also show for all $2<p,q<\infty$ that there is a closed
subspace $X\subset\ell^q$ that fails the $\mathcal{SK}_r$-approximation
property for all $r\ge p$.
