detected. A circle of radius 0.3 arcsec centered at the *Chandra* position is plotted to ease the visualization of the distance between the ULX and the radio components. The beam size shown at the bottom left corner is 33 mas $\times$ 27 mas oriented at a position angle of 86$^{\circ}$.4.\[radio\]](N4088casa2.pdf)

In the 5 GHz EVN observations, no radio emission is detected above a 5$\sigma$ level for N4088–X1 within 1 arcsec of the *Chandra* position nor within a FOV of 5 arcsec around the VLA position. An upper limit on the flux density of the ULX of 0.30 mJy beam$^{-1}$ is obtained by estimating the rms at the *Chandra* position, from which we derive an upper limit on the brightness temperature $T_\mathrm{B} < 6 \times 10^{5}$ K. Adopting a distance to NGC4088 of 13 Mpc, we derive an upper limit on the 5 GHz radio luminosity of $L_\mathrm{5 GHz} < 3.1 \times 10^{35}$ erg s$^{-1}$.

Combined with the 1.6 GHz detection, the upper limit on the flux density at 5 GHz can be used to constrain the spectral index of the source. Defining $F_{\nu}\propto\nu^{\alpha}$, we obtain $\alpha\leq1.6$, which is trivially satisfied by any physical class of radio spectra. It should be 