Rakshasa’s Limited Magic Immunity – does it extend to spells that are ongoing on it?

A Rakshasa has the following trait:

Limited Magic Immunity. The rakshasa can’t be affected or detected by spells of 6th level or lower unless it wishes to be. It has advantage on saving throws against all other spells and magical effects.

Rakshasas can cast Disguise Self at will.

If a Rakshasa has cast Disguise Self on themselves (choosing to have it effect them), can a PC who has cast Detect Magic see the effect of the Disguise Self spell?

This question can be rephrased in three different ways (which may be useful as a way to consider this effect):

  • Does the PC see that the Rakshasa has the effect of an illusion spell on it?
  • Does the Rakshasa’s Limited Magic Immunity extend to spells that have been cast on it and thus mask the Detect Magic?
  • Does the fact that the Detect Magic spell is cast on the PC (changing their sight) bypass the Limited Magic Immunity of the Rakshasa?

Note: The answer to this question will apply more generally to other ongoing effects, but for the purpose of getting a succinct answer I am restricting this question to cover the interaction between Detect Magic and Disguise Self as they pertain to a Rakshasa.

For reference Detect Magic has this text:

For the duration, you sense the presence of magic within 30 feet of you. If you sense magic in this way, you can use your action to see a faint aura around any visible creature or object in the area that bears magic, and you learn its school of magic, if any.

The spell can penetrate most barriers, but it is blocked by 1 foot of stone, 1 inch of common metal, a thin sheet of lead, or 3 feet of wood or dirt.

and Disguise Self has this text:

You make yourself–including your clothing, armor, weapons, and other belongings on your person–look different until the spell ends or until you use your action to dismiss it. You can seem 1 foot shorter or taller and can appear thin, fat, or in between. You can”t change your body type, so you must adopt a form that has the same basic arrangement of limbs. Otherwise, the extent of the illusion is up to you.

The changes wrought by this spell fail to hold up to physical inspection. For example, if you use this spell to add a hat to your outfit, objects pass through the hat, and anyone who touches it would feel nothing or would feel your head and hair. If you use this spell to appear thinner than you are, the hand of someone who reaches out to touch you would bump into you while it was seemingly still in midair.

To discern that you are disguised, a creature can use its action to inspect your appearance and must succeed on an Intelligence (Investigation) check against your spell save DC.