Approximation properties $\mathrm{AP}_s$ and $p$-nuclear operators (the case where $0<s<1$)

Among other things, it is shown that there exist Banach spaces $Z$ and $W$ such that $Z^{**}$ and $W$ have bases, and for every $p\in[1,2)$ there is an operator $T\colon W\to Z$ that is not $p$-nuclear but $T^{**}$ is $p$-nuclear.