Ivory Soap
I can see with the addition of STPP and use of Ivory Soap, that acceptable results could be achieved. However it's important to point out the lye concentration I mentioned up thread. Since Ivory is in fact made to be used on skin, it means that at the very least, the amount of NaOH used, is only what's required to produce a bar suitable for bathing...and no more. Last time I looked at Ivory, the glycerin remains intact.
For a true laundry soap, all fat and or oils should be reacted, (slightly over reacted would be better) then during processing the raw soap should be purified with a salt solution. This causes the glycerin (a by product of soap) to be extracted, along with some other impurities (spent lye) to be removed. What's left is a pure or purified soap. Additives such as STPP, Borax, Soda, carbolic acid, stoddard solvents, bluing agents (the list can go on) can be added to assist or customize the soap. Just those two adjustments (reacting all fat, and purifying) can make huge difference in how the same basic formula will perform for laundry use.
To my knowledge there is only one manufacture that takes the time to do the first two steps mentioned. Zote, Fels, and Ivory are not taking those steps. At least not these days. And while I have used them for one thing or another, I have never considered them to be "pure laundry soap" but that just me.
Without going into a lot detail, its worth noting that each fat or oil contains its own fatty acid make up and no two are exactly alike. Those differences can be used to the soap makers advantage. The point I'm making here, is that the selection of the fats used to start with, play a part in how a laundry soap will clean, and rinse from fabric.
When you take into consideration all of this, it's not much of a stretch to believe in the possibilities.