Have you thought about using winders? With a 6' opening , you will need to use a 6 piece winder for a complete u-turn. Example: starting from 3rd floor, you will have 3 steps, then 6 winding steps at the wall, then a final 3 step to 4th floor.

You need a minimum 13 risers for a 9' span. Min tread width is 10 inches within the clear width of the stair to meet the code, that will put you at 130" which is a 10'10" run. There is no way to fit that stairway into 6' opening.
In order to make this work you have to make your stair tread with 5 1/8", so you might as well put a ladder there.

planning to do this
what do you think

planning to do this
what do you think

I think it looks good on the paper but to make a code compliant staircase you will have to call a stair company have them do the layout and they will tell you if you have room or not and give you what options you have.

In here you wasting your time.

With that said, they can probably fit 3' landing with 2 stairs going down and doing 90 degree turn with 10 stairs, that will set you at around 8"+ riser height, here in NJ you can have a riser height 8 1/4" and min thread depth 9" so it might just work.
You have to check your local code and see what the stair requirement is for your area and work from there.

Good luck

Spiral would definitely work and you can have a nice size spiral staircase with the space he has, but he said he don't want that.

You also have to be careful with spirals they are not always up to code without an other way of egress where I live you can only have a spiral to a second floor if you also have an outdoor set of stairs for emergency crews.

6 straight treads , 6 winders and 1 nosing will give you 13 rise. Tread depth at 10" and width 36". You can fit this set in a 6'x6' opening.