We present the discovery of the fourth persistent radio source (PRS) associated with a fast radio burst (FRB). Following previous indications of a candidate PRS associated with FRB20240114A, we performed deep VLBA observations at 5 GHz to test the presence of a compact radio source within the uncertainty position of this FRB ($\pm$200 mas). We detect a component $\sim$50 mas northwards the nominal position provided by the PRECISE collaboration. The corresponding radio luminosity, together with the Faraday rotation measure provided by previous observations of the FRB, locate this PRS in the expected region of the $L$ vs |RM| relation for the nebular model, further supporting it. The comparison of the measured flux density with the values collected at lower frequency by previous studies indicates a possible steepening of the radio spectrum in the 1-5 GHz range. Optical observations performed with the LBT could reveal that the FRB and its PRS lie at $\sim$1 kpc from the centre of the host galaxy, which is a dwarf sub-solar metallicity starburst galaxy with SFR $\sim 1 M_\odot\;\mathrm{yr^{-1}}$ and stellar mass $M\sim10^8 M_\odot$.

At the dawn of the gamma-ray burst (GRB) afterglow era, a Cepheid-like correlation was discovered between time variability V and isotropic-equivalent peak luminosity Liso of the prompt emission of about a dozen long GRBs with measured redshift available at that time. Soon afterwards, the correlation was confirmed against a sample of about 30 GRBs, despite being affected by significant scatter. Unlike the minimum variability timescale (MVT), V measures the relative power of short-to-intermediate timescales. We aim to test the correlation using about two hundred long GRBs with spectroscopically measured redshift, detected by Swift, Fermi, and Konus/WIND, for which both observables can be accurately estimated. For all the selected GRBs, variability was calculated according to the original definition using the 64-ms background-subtracted light curves of Swift/BAT (Fermi/GBM) in the 15-150 (8-900) keV energy passband. Peak luminosities were either taken from literature or derived from modelling broad-band spectra acquired with either Konus/WIND or Fermi/GBM. The statistical significance of the correlation has weakened to <~2%, mostly due to the appearance of a number of smooth and luminous GRBs characterised by a relatively small V. At odds with most long GRBs, 3 out of 4 long-duration merger candidates have high V and low Liso. Luminosity is more tightly connected with shortest timescales measured by MVT rather than short-to-intermediate ones, measured by V. We discuss the implications on internal dissipation models and the role of the e+- photosphere. We identified a few, smooth GRBs with a single broad pulse and low V, that might have an external shock origin, in contrast with most GRBs. The combination of high variability (V>~0.1), low luminosity (Liso<~10^51 erg s^-1) and short MVT (<~ 0.1 s) could be a good indicator for a compact binary merger origin.