In this work, which is a continuation of Castelló-Mor et al. (2016), we present new X-ray and infrared (IR) data for a sample of active galactic nuclei (AGN) covering a wide range in Eddington ratio over a small luminosity range. In particular, we rigorously explore the dependence of the optical-to-X-ray spectral index $α_{OX}$ and the IR-to-optical spectral index on the dimensionless accretion rate, $\dot{\mathcal{M}}=\dot{m}/η$ where $\dot{m}=L_{AGN}/L_{Edd}$ and $η$ is the mass-to-radiation conversion efficiency, in low and high accretion rate sources. We find that the SED of the faster accreting sources are surprisingly similar to those from the comparison sample of sources with lower accretion rate. In particular: i) the optical-to-UV AGN SED of slow and fast accreting AGN can be fitted with thin AD models. ii) The value of $α_{OX}$ is very similar in slow and fast accreting systems up to a dimensionless accretion rate $\dot{\mathcal{M}}_{c}∼$10. We only find a correlation between $α_{OX}$ and $\dot{\mathcal{M}}$ for sources with $\dot{\mathcal{M}} > \dot{\mathcal{M}}_{c}$. In such cases, the faster accreting sources appear to have systematically larger $α_{OX}$ values. iii) We also find that the torus in the faster accreting systems seems to be less efficient in reprocessing the primary AGN radiation having lower IR-to-optical spectral slopes. These findings, failing to recover the predicted differences between the SEDs of slim and thin ADs within the observed spectral window, suggest that additional physical processes or very special geometry act to reduce the extreme UV radiation in fast accreting AGN. This may be related to photon trapping, strong winds, and perhaps other yet unknown physical processes. ; Comment: Accepted for publication on MNRAS, 14 pages, 7 figures