  We show that the momentum-dependent scaling exponents of the holographic
fermion self-energy of the conformal-to-AdS$_2$ Gubser-Rocha model can describe
new findings from angle-resolved photoemission spectroscopy experiments on a
single layer (Pb,Bi)$_{2}$Sr$_{2-x}$La$_x$CuO$_{6+\delta}$ copper-oxide. In
particular, it was recently observed, in high-precision measurements on
constant energy cuts along the nodal direction, that the spectral function
departs from the Lorentzian line shape that is expected from the
power-law-liquid model of a nodal self-energy, with an imaginary part
featureless in momentum as $\Sigma''_{\text{PLL}}(\omega) \propto
(\omega^2)^\alpha$. By direct comparison with experimental results, we provide
evidence that this departure from either a Fermi liquid or the power-law
liquid, resulting in an asymmetry of the spectral function as a function of
momentum around the central peak, is captured at low temperature and all
dopings by a semi-holographic model that predicts a momentum-dependent scaling
exponent in the electron self-energy as $\Sigma(\omega,k) \propto \omega
(-\omega^2)^{\alpha (1 - (k - k_F)/k_F) - 1/2}$, with $\hbar k_F$ the Fermi
momentum.
