Credit goes to Flakmeister of wowhead.com

It's going to be one beast of an achievement, that's for sure. In fact, I have difficulty seeing it being possible at the moment.
Aside from the fact that you'll have to do without any statistical bonuses granted by the Keepers (40% extra damage, 20% more HP, 20% additional Healing taken, 20% speed increase and a 20% reduction in damage taken) you will also have to work around the lack of:

1: Hodir's Protective Gaze, this is pretty much a "get-out-of-jail-free" card which isn't really needed if the tanks are up to snuff.

2: Mimiron's Destabilization Matrix. Things are getting a bit more tricky now. Considering that one half of your DPS is diverted to the Brain during phase 2 you need one heck of a DPS and Healing team to both dispose of the tencacles before they overwhelm the raid and heal through the high damage output which they produce. This can also turn things even more grim when you factor in the lack of the following ability.

3: Thorim's Titanic Storm which is used for killing the adds during phase 3. Without it, Yoggi will be able to heal his minions back up when he uses Empowering Shadows meaning you're looking at waves of constantly spawning Immortals that will mess up your tanks good, if the DPS doesn't burn Yoggi down in an expedient manner.

4: Add to that this last twist. Freya won't be there to give you Sanity Well, making this encounter specific resource one you cannot afford to squander. It's also probably the last safety measure your raid will want to disable, as unlucky RNG can cause raid members to be useless because Sara chose to cast her vast assortment of psychosis related spells on members a consecutive number of times.

In summation: Doing the encounter without 1-2 Keepers up should represent a fairly feasible challenge. When you get to 3, the difficulty increases exponentially and with Alone in the Darkness the function representing difficulty turns into a logarithmic one. (I'm not sure the mathematical metaphor stands. Math is yet another field of classical education in which I fail.)