The `1/9'-problem for the $φ$-functions

The functions $φ_\ell(z)=(e^z-s_\ell(z))/z^\ell$, with $s_\ell$ the Taylor polynomial of $e^z$ of degree $\ell-1$, are what exponential integrators evaluate, and they evaluate them through rational approximations on the negative real axis $\mathbb{R}_-$. Schmelzer and Trefethen (2007/08) conjectured