## 2 Experimental Section  
The 3 rd-order IVSTF-NLE used in this experimental study is based on the algorithm published in [16]. For completeness and for the reader's convenience, we summarize below the major schematics highlighting its operating principle, whereas the theoretical model is summarized in the Appendix A. The block diagram of the 3 rd-order IVSTF-NLE is shown in Figure 1a. The equalization is performed in parallel. In the block diagram of Figure 1a, the linear compensation is represented by the two upper arms and the nonlinear compensation by the \(N\) nonlinear branches. The linear compensation comprises a single branch per polarization, accounting for the chromatic dispersion (CD) introduced by all the \(N\) fiber spans, in the frequency domain. Each polarization tributary passes through a filter with transfer function \(K_{1}(\omega)\) (see Expression (A5) in the Appendix A). On the other hand, the nonlinear compensation is implemented in a span-by-span basis through \(N\) separate branches, as described in (A6). Figure 1b illustrates the operating principle of the \(k\)th nonlinear branch, which is divided into three stages. The first stage performs CD compensation in the frequency domain, and it is realized through the filter \(\left(K(\omega)\right)^{k}\) (see Expression (A7) in the Appendix A). The second stage is the compensation of the intra-channel nonlinearities, in the time domain, implemented as described in (A8). The third and the last stage of each nonlinear branch is the compensation of the residual dispersion, in the frequency domain, through a filter with transfer function \(\left(K(\omega)\right)^{N-k}\) (see Expression (A9) in the Appendix A). More specifically, first, the received \(x\) and \(y\) polarization tributaries (i.e., \(A_{x}\) and \(A_{y}\)) are linearly compensated in the frequency domain. Then, the linearly-compensated polarization tributaries are switched back in the time domain (i.e., \(A_{x}^{LE}\) and \(A_{y}^{LE}\)) for the intra-channel nonlinearity equalization where each one of them is multiplied with the instantaneous power of \(x\) and \(y\) polarization (i.e., \(\left|A_{x}^{LE}\right|^{2}+\left|A_{y}^{LE}\right|^{2}\)) scaled by the adjustable parameter \(C_{IVSTF}\). The optimization of the latter is achieved by sweeping a range of values in the vicinity of its nominal value \(C_{nominal}=(8/9)\gamma L_{effr}\)  
Figure 1: **(a)** Block diagram of the 3 rd-order IVSTF-NLE; **(b)** Operating principle of the kth nonlinear compensation branch of the 3 rd-order IVSTF-NLE [16]. (Symbols: Inverse Fast Fourier Transform (IFFT), Linearly Equalized (LE), Non-Linearly Equalized (NLE), Fast Fourier Transform (FFT)). where \(L_{eff}=\left(1-e^{-\alpha L}\right)/\alpha\) is the effective span length. Finally, the polarization tributaries \(A_{x}^{LE,NLE}\) and \(A_{y}^{LE,NLE}\) are switched again to the frequency domain, where the residual CD is compensated through a filter with transfer function \(\left(K(\omega)\right)^{N-k}\). The experimental setup is depicted in Figure 2. The configuration consists of a coherent, DP 16 QAM OFDM super-channel in a 10 \(\times\) 100 km transmission line carrying a net bit rate of 400 Gb/s. The OFDM super-channel consists of 4 bands. Thus, the net bit rate per band is equal to 100 Gb/s. By adding 20% soft-decision forward error correction (SD-FEC) overhead and 20% OFDM overhead, including the cyclic prefix (CP), as well as the training sequence (TS) and the pilot tones, then, the raw bit rate per band is 144 Gb/s. Each band occupies 18 GHz (Figure 3d). A guard-band of 2 GHz is introduced to limit the crosstalk between adjacent bands. The total bandwidth of the super-channel is 78 GHz, corresponding to a net spectral efficiency of 5.13 b/s/Hz. The OFDM frame is generated by Matlab(r) using a 1024-point fast Fourier transform (FFT)/inverse FFT (IFFT). There are 576 subcarriers, each carrying 16 QAM. They are generated by a 15-GHz bandwidth Keysight(r) arbitrary waveform generator (AWG) operating at 64 Gsamples/s.