Here is what I was thinking:-
The thread would not need to be run up the channel but behind it, that clear fishing stuff connected to the bottom edge of the bottom, the 3rd one from bottom and 5th from bottom doors, a thin piece of thin plasticard would hide the threads behind it at the wall.
The threads would be under tension to pull the bottom edge inside, then they can be loose for the remainder of travel, set back from the inside channel is another thread that is on a sail winch type mechanism that travels from the bottom door to pull up to the where the bottom is flat at the top of travel, it is pulling sightly towards the inside - but due to the travel of the doors it cannot pull inside, the loop of the fishing line goes over a solid pulley / pin at the top (thats where this tension comes from if the pin is set back from the door), the lifting mechanism is under the deck as there is no space in the upper structure above the door.
The doors would need to be made of brass as it would not be long before the test item shown would fall apart or wear away, there would be a solid pulley / pin at the base as well, the only issue is a bar that would move into position at the other side of the door to push it back to being flat and therefore put the lines at the bases of the door under tension again.
The concept is in my head, the issue is drawing it to scale to see if it would work and how to make it at this scale.
What popped into my head was three designs that are out there they would need investigating to provide possible linkage designs.
1st who remembers the old school louvered windows - where pulling a lever down or up the glass panes would move at the same angle to be opened the issue was linking an opposite angled window and pushing them up.
2nd Louvered blinds - again the issue was it being pulled up and down, the same concept off the bottom door can be employed, its remembering that the centre is inside the door not on its travel in the channel pins, also space constrains mean any motors need to be under the deck.
3rd the solid linkage, its like the lock on a filing cabinet, a flat bar runner up the back of the doors slide with little ledges, as its pushed up, little corresponding ledges on the doors that dont interfere with each other are pushed up by the lifter - the upper two could only make contact when the doors need moving from the vertical, the bottom one could push up the bottom and the others up as one, ideally the upper two from one side and the bottom on the other side - why, the two upper ones would interfere with the roof as they got closer.
You have to see the images in my head