The paper mentions (page 117) that there are accuracy concerns with the FT quadrature method used for equation 8.21 (and analogous ones for others) for "near maturity deep OTM and ITM calls (puts)". This is attributed to the highly oscillatory nature of the Fourier Transform kernel. Is performance important here? Like, would a better quadrature rule which didn't suffer due to the oscillatory kernel help in an important way?

I ask, of course, because there are better ways to treat Fourier-type integrals than trapezoidal rule (I even wrote a numerical analysis paper on such a method).