By definition a major piece is one that can (together with its own King) force checkmate against a bare King from almost any position. (There could be a few tactical positions where the piece gets lost or stalemate is unavoidable.) In orthodox Chess the difference between major (Queen, Rook) and minor (Bishop, Knight) pieces nicely corresponds with piece value (light vs. heavy pieces). In general this need not be the case, however. The ability to force checkmate on a bare King is a very specific one, which only correlates poorly with the general power of a piece. And the value of pieces in chess is mainly determined by how well they support or combat Pawns, as promotion of the latter is usually decisive.
So there can be very valuable minor pieces. E.g. consider a piece that is allowed to leap directly to every square of the board that has same shade as the one it is on. It can never inflict checkmate, not even against a cooperating opponent ('helpmate'), but in the middle game it is more powerful than a Queen. At the other end of the spectrum, a Rook that is restricted to move at most two squares (and thus can reach at most 8 squares in total, the access to half of those can be blocked, and is worth less than a Knight, which can unconditionally reach all its 8 targets), which still has mating potential. There even are major pieces with only 5 moves, hardly worth more than 2 Pawns in the middle game.
From orthodox Chess we know that being a minor ahead in a pawnless end-game is usually not enough to force a win. (While being a Pawn ahead often is a winning advantage.) The same holds for being an 'exchange' ahead, i.e. having one significantly more powerful piece in an otherwise equal (but pawnless) position, provided the value difference is not more than that of a light piece (B or N). A 'super-piece' like Queen will beat all other pieces in a 1-to-1 situation, though.
Musketeer Chess complicates this picture by adding many piece types that are between Rook and Queen in value, and even one (Dragon) that is far stronger than a Queen. Some of those definitely classify as super-pieces, having a value very close to that of a Queen. The value difference between the Musketeer pieces is in general not large enough for one to beat the other in 'single combat', with the exception of the Dragon: the latter combines the powers of Queen and Knight, not enough to beat the Queen and nearly equivalent Chancellor, but good enough to beat all other Musketeer pieces.
The Musketeer Unicorn even is a minor piece, albeit more valuable than a Rook, so it could not even beat a bare King, let alone a supported one. It turns out the Hawk and Unicorn are significantly weaker than the others, and definitely are not superpieces; they cannot subdue even an opposing Bishop or Knight. A Rook is a pretty tough defender, btw, and that a Queen (and thus a Dragon) can beat it is really an exception: no other Musketeer piece can do that. Hawk and Unicorn do much worse as defenders, and loose against the stronger Musketeer pieces Chancellor, Spider and Cannon as well as against Queen/Dragon.
I summarized the 1:1 and bare-King results in the following table, where '+' means the end-game is generally won by the piece on the left, and '-' means it is a general draw (or worse, for the defender). Pieces are given as the first letter of their usual name, except that I used M for Chancellor, because this piece is also known under the name 'Marshal', and C was needed for the Cannon. A '?' means a result that is unclear from the statistics: significantly fewer lost positions, but far more than for a general draw. Probably this means an end-game where in some fortress position for the defender exists from which he can hold out indefinitely, if he succeeds in reaching it. (Could also be a perpetual check.)
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1 vs 1 defender D Q M S C A E L F H U R B N - D - - - ? + ? + + + + + + + + + Q - - - - - - - - + + + + + + M - - - - - - - + + - + + + S - - - - - - - - + + - + + + C - - - - - - ? ? + + - + + + A - - - - - - - - - + + + E - - - - - - - - - + + + L - - - - - - - - - + + + F - - - - - - - - - - + + H - - - - - - - - - - - + U - - - - - - - - - - - -
them. (Except perhaps in the case Spider + Knight vs Spider, where a forced win does in general exist if it were
not for the 50-move rule.) This because after trading the stong pieces, the remaining minor cannot win. Even a small additional advantage of the side with the extra minor is enough to tip the balance in his favor, though.
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extra Knight defender D Q M S C A E L F H U R DN - + + + * + * * * * * * QN - - ? + ? + + + * * * MN - ? + ? + + + * * + SN - - - ? ? + + + * * + CN - - - - - + ? ? * * ? AN - - - - - 5 + + ? EN - - - - ~ - + + ? LN - - - - - - + + ? FN - - - - - - + + ? HN - - - - - - - - UN - - - - - - - - RN - - - - - - - - extra Bishop defender D Q M S C A E L F H U R DB - ? ? + * + * * * * * * QB - - ? + + + + + * * * MB - ? + ? + + + * * + SB - ? ? + + + + * * + CB - - - - ~ - + + * * + AB - - - - - - - - + + + EB - - - ? 5 + + ? LB - - - - - + + ? FB - - - - - - + + ? HB - - - - UB - - - RB - - -