Thanks for the screenshots. That’s exactly as expected and it’s caused by the linear phase LF rolloff in the filtered loudspeaker response. The only way to have a step response that steps up and stays up is to have a loudspeaker that has a flat response to DC.
To investigate the effect more, load FIR Designer without any loudspeaker measurement on the Import tab, then put a HPF filter either on the IIR filters tab or the Mag Adjust tab. If using the Mag Adjust tab, try minimum and linear phase and watch the effect on the step response.