Yes under normal circumstances the board can work fine at 28V or 24V. This is a consistent issue related to a momentary power disconnect followed by full power again (around 50ms time disconnected or less).
Based on the datasheet, the SBC seems to enter the DEEP-FS state after the power interrupt. Then the infinite loop begins. VPRE and all of the regulators go off for approximately 4s and then come back on for about 1s over and over. It appears that the infinite autoretry OTP feature is occurring while in the DEEP-FS. WAKE1 and VBOS are both on the whole time in the error state, even when VPRE and the regulators turn off.
What I am not sure of is why, once power is steady, VPRE and the regulators continue to turn off. Remember, this issue only occurs when power is supplied, removed, and quickly reapplied (within far less than a second). If power is turned on normally, my design works fine.
In case it helps, I've linked an image of my schematic.