The fuzzy phase spreading could be ionospheric "multipath". If you have a stable LO frequency reference (GPSDO) you can watch WWV shift phase (frequency is the integral of phase shift w.r.t. time) shortly after sunrise and sunset as the path length changes. The maximum deviation I have measured was 2 hz (10 mhz WWV) but there are papers in the IRE Proceedings that show graphs of up to 5 hz short term peak deviation. As I had used 10 Mhz WWV as my frequency reference to set my AIL, IFR, Cushman, and Agilent test sets (1960's to 2013) it was a bit jarring to think about the potential "error" simply because I was too "cheap" to invest in a WWVB 60 khz receiver. . .
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FSQ-2 is related to WSQ. On the WSQ page you will find the following note:
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Don't even think about using the software on HF. There is far too much Doppler shift (except on ground wave) for the receiver to work reliably. Even 160m will be a challenge at night.
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Interesting comment. . .? Seems so true about all of HF radio. Sometimes the ionosphere helps and sometimes it doesn't.
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JIm/VEZ
James Redding WA9VEZ