Can someone explain how this code fulfils the HPS (harmonic product spectrum) algorithm?

Question

For a project, I'm working on methods of pitch detection and have settled on the idea of using Harmonic product spectrums. I've been reading up on the theory and as I have no background in music, I'm finding it to be a bit tricky to understand. I'm using processing with the minim library and stumbled across this code online. I understand everything up until the downsampling part. I'm aware that one has to apply a hanning window, then fft, then downsample and multiply by compressing the sample. I know this in theory, but I'm struggling to understand how the code below does it. If possible could someone shed light on how exactly this algorithm implements that so I can adapt it and actually understand what I'm doing. Thanks and I appreciate any feedback.

class PitchDetectorHPS{

  FFT fft;
  int sampleRate;
  int fftLength;
  int harmonicSize;
  float[][] step;
  PitchDetectorHPS(int fftLength,int sampleRate,int harmonicSize){
    fft=new FFT(fftLength,sampleRate);
    this.sampleRate=sampleRate;
    //this.fftLength=fftLength;

    this.fftLength=fft.specSize();
    this.harmonicSize=harmonicSize;
    step=new float[harmonicSize][];
    for(int i=0;i<harmonicSize;i++){
      step[i]=new float[this.fftLength];
    }
   //fft.window(fft.HAMMING);
  }
  float detect(float[] frame){
    fft.forward(frame);
    //downsample
      for(int i=0;i<fftLength;i++){
        for(int j=1;j<=harmonicSize;j++){
           if(i%j==0)( step[j-1])[i/j]=fft.getBand(i);
        }
      }
     //HSP
      int index=0;
      float max;
      max=0;
      float tmp;;
      for(int i=0;i<fftLength;i++){
        tmp=1;
        for(int j=0;j<harmonicSize;j++){
          tmp*=step[j][i];
        }
        //println(tmp);
        if(tmp>max){
          max=tmp;
          index=i;
        }
      }
       if(index==0)return 1.0;
      return index*fft.getBandWidth();

  }
}

ETA: I've actually tried to implement it but am getting repeated values of 86.13281 when using an electronic song. I'm guessing this isn't right. When using a series of flute notes, I get correct frequencies initIally, but never a frequency above 1K.

ETA: Ok so I've been experimenting some more. I tested it using this video https://www.youtube.com/watch?v=d7lJ_nyCDQA

视频上的频率是 602,689，732,818,904,990,1119，2369

我得到的频率是

613、689、732、807、915。1022、1098，并且不会超出此范围。直到更高的频率，一切都还可以。谁能提供一些见识？