The MPT algo I'm using right now is the best I've heard in retaining the overall tone
There should be no differences in MPT "algorithms". MPT is not an algorithm. It is a mathematical function. It is the Hilbert Transform of the log of the magnitude. There is one and only one minimum phase transform of a signal. If you hear a difference between two "algorithms" then one (or both) of them isn't implementing the function properly.
Furthermore speaker drivers are minimum phase by nature (1). Tweeters can exhibit some non-minimum phase behavior but for guitar speakers the response is essentially minimum phase. Crossover networks are typically not minimum phase but guitar speakers don't use them. Multiple speakers (i.e. 4x12) may not be minimum phase if captured far-field off-angle. Anyone claiming to hear a difference in close-mic'd IRs is fooling themselves.
If you compare a truncated "raw" IR with a truncated MP IR there may be differences in the low frequency response which get more pronounced the more you shorten the IR. This is because the MP IR "preserves" LF information in the truncated version since it "pulls" the basis functions at those frequencies towards the start. If there is a marked difference then this indicates there is considerable room content in the IR. FWIW, I have not measured significant deviations between normal and MP in the OwnHammer IRs.
MP solves a lot of problems and is inaudible when implemented properly. Humans are very insensitive to phase. Any minor differences in phase between the raw and MP versions would be inaudible. What IS audible is the truncation.
(1) "Most loudspeaker drive units, excluding the crossover networks, exhibit a minimum-phase frequency response"
Digital Signal Processing Tools for Loudspeaker Evaluation and Discrete-Time Crossover Design
MALCOLM OMAR HAWKSFORD, AES Fellow
Centre for Audio Research and Engineering, Department of Electronic Systems Engineering,
University of Essex, Colchester C04 3SQ, UK