------------------------ ---------------------------------------- ------------------------------------------- -------------- ---------
  Model          $N_\mathrm{H}$       $\Gamma/T_\mathrm{in}$                   Norm.                                      Flux                      $\chi^2$/dof   p-value
              (10$^{22}$ cm$^{-2}$)          ( /keV)                                                            (erg cm$^{-2}$ s$^{-1}$)                          
  *pow*        0.3$^{+0.5}_{-0.3}$     1.1$^{+0.6}_{-0.5}$     (1.6$^{+2}_{-0.7}$) $\times$ 10$^{-5}$   (1.7$^{+0.6}_{-1.5}$) $\times$ 10$^{-13}$     16.9/10       0.08
  *diskbb*     0.2$^{+0.3}_{-0.2}$      2.5$^{+4}_{-0.9}$              2.1 $\times$ 10$^{-4}$           (1.4$^{+0.2}_{-1.4}$) $\times$ 10$^{-13}$     15.62/10      0.11
  ---------- ----------------------- ------------------------ ---------------------------------------- ------------------------------------------- -------------- ---------

### *Swift* ULX spectral modeling

We fitted the co-added *Swift* spectrum using [XSPEC]{} again, with an absorbed power-law model with absorption accounted for using the *wabs* model and the $N_\mathrm{H}$ column set to be greater than the Galactic absorption in the direction of N4088–X1 (i.e. 2 $\times$ 10$^{20}$ atoms cm$^{-2}$). The fit obtained, with a hard photon index of 1.3 (see Table \[specfit\]), does not satisfactorily describe the observed spectrum, as two data points above 5 keV fall below the power-law fit (see Fig. \[figSwift\], top). We thus attempted to fit this spectrum with *wabs\*diskbb* as with our *Chandra* data and an exponential cut-off *wabs\*pow\*highe*. Using the *wabs\*diskbb* model, a fit with $\chi^{2}$/dof=39.97/34 and an inner disk temperature of 2.1 keV (see Table \[specfit\]) is obtained. This temperature is again too high for optically thick emission from a typical optically thin accretion disk. A good fit ($\chi^{2}$/dof=36.89/32) is also obtained with the 