The PSE waveform looks familiar, but the voltage scale of 1kV does not see correct. If this is a true waveform, the timing at the PSE leads me to believe there is a DC disconnect at the PSE.
What is the output of the PD connected to? The PD must draw a DC MPS minimum current to remain on. The LT4293 demo board sets the LED current limiting resistor to draw this minimum load; But the LED load is for demo purposes since the demo board output is open for connection to the application load. The PD schematic show a much larger resistor value who reduces the load current below the DC MPS if JP3 is not connected to anything. The PD must draw more than 10mA to remain on.