  Resonant optical excitation of certain molecular vibrations in
$\kappa$-(BEDT-TTF)$_2$Cu[N(CN)$_2$]Br has been shown to induce transient
superconducting-like optical properties at temperatures far above equilibrium
$T_c$. Here, we report experiments across the bandwidth-tuned phase diagram of
this class of materials, and study the Mott insulator
$\kappa$-(BEDT-TTF)$_2$Cu[N(CN)$_2$]Cl and the metallic compound
$\kappa$-(BEDT-TTF)$_2$Cu(NCS)$_2$. We find non-equilibrium photoinduced
superconductivity only in $\kappa$-(BEDT-TTF)$_2$Cu[N(CN)$_2$]Br, indicating
that the proximity to the Mott insulating phase and possibly the presence of
preexisting superconducting fluctuations are pre-requisites for this effect.
