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Ix'mnk - soloable by a BLM SCH or RDM
Plan: Kite to circle tower room, die to lose adds, RR up and kite whilst weakened. Use Bio, Drain and helices (if sch) for damage. Must have some form of innate refresh e.g. vcloak. For more info, see "Frodo FFXI BLM Solo Ix'aern MNK" on YouTube.
Ix'drg - soloable by BLM SCH or RDM but a lot harder
Plan: Kill PHs to pop it (hardest part). Stick on dots then zone, rinse and repeat. Must time dots so there is always something on it at all times else it will regen.
Ix'drk - duoable by BLM SCH RDM, soloable by RDM with exceptional gear
Plan: One takes adds to zone, one takes ix'drk to circle tower room and kites, use dots, light nukes also acceptable. RDM can solo this because they can chainspell Gravity the adds and zone the adds and the NM, but with a big gap between adds and NM, allowing the NM to be claimed after you zone back in, without the adds linking. A similar method could probably be employed using Manifestation+Gravity, but it would be riskier.
Jailer of Fortitude - Soloable by BLM SCH RDM COR
Plan: Movement speed+ is required by default. I don't know how well Desert Boots will work in substitution. Pop it, kite the jailer around ru'hmet, adds will eventually lose aggro but the jailer will still have reflect up. Bring jailer to circle tower room and kite, using dots for damage, /nin is required to avoid getting hit. Little need for /rdm anyway since its immune to grav/bind/sleep.
Jailer of Faith - Possibly duoable by SCH + [BLM RDM SCH]
Plan: Movement speed+ is required by default. Kite to the circle tower room and keep kiting, ensure the sch is out of pt for stona, stick on grav when things look bad, use dots. Running using +12% can avoid most of the jailer's hits and spells, due to him casting frequently. Alternatively, take it to the teleporter zone and possibly solo it there, treating it like Ix'drg. Beware of Breakga.
Jailer of Temperance - Rumored to be soloed by two RDM/NINs
Plan: Stand either side of the circle tower room and let the jailer bounce between you, i don't know much info on this.
As BLM, I confirm the following:
Ix'MNK;
Ix'DRG;
Ix'DRK; and
Jailer of Fortitude
I did not try myself Faith and Temperance, tho I have seen Temp's strategy being executed. Apparently what they did was DoT and grav, then outran him until they could position themselves in the opposite pillar from the room. They changed direction constantly, which made the jailer change as well.
On an additional note, Ix'mnk can be solo killed during night time by a NIN/RDM with some specific gear (I think it's Koga mask).