Here goes:
This is a free body force diagram on any particular chunk of mylar:
Thin films can only exert forces in the plane and not perpendicular to it. The two horizontal tension forces sum into a net force that pulls the chunk down into the rear stator.
Nothing opposes this force, which is why I said that "it is theoretically impossible for a curved...."
The way to maintain this shape is to have zero horizontal tension.
But if you DO have some horizontal tension, the only way to oppose this force is to have a vertical tension. This pulls the mylar chunk into a saddle shape, with two edges pulled downwards, and two pulled upwards:
The "angle" that the force pulls at is approximately equal to the inverse radius of curvature in the horizontal or vertical direction. So the final force balance equation is Tv/Rv=Th/Rh. Usually the radius of curvature in the vertical direction is very very large since the panel is flat in that direction. To get the equation to balance that means you either need a very very large vertical tension, or a very very small horizontal tension.
For now, suppose the vertical and horizontal tensions are equal. (I'm going to show why this is wrong.) The radius of curvature in the horizontal direction is... umm, I dunno, but for a 9" wide panel, the edges are recessed by about 1" or so. In the vertical direction, take a spar spacing of 3". For the same radius of curvature (required if the tensions are the same), the droop is approx the square of the length of the segment. So (3/9)^2=1/9". Nope that would jam the film into the into the rear stator. The horizontal tension has got to be wayyyy less.
The ML has a stator spacing of... I dunno, it was either .04" or .06" but in any case, a reasonable upper bound for the sag might be around .01" or so between the stators. That works out to about 12x less horizontal tension than vertical in this case (using very approximate math). So check your own speakers to see how much sag you have between the spars and that will tell you how much horizontal film tension you have. Whatever it is, it is still a very small fraction (or none if the film is totally flat) of the vertical tension.
You can check the total tension by lightly poking at the film with an insulating stick and see how much it deflects. On mine, it deflects fairly easily, so I know the total tension is low. That means the horiz tension is very very low.
As to the videos showing them laying out the film. Yup, you can see the sag between the spars as they lay it out. But then they take the panel out of the jig. The horizontal tension is going to cause the perforated sheet metal to unbend slightly, while the vertical tension mostly remains because the sheet metal is really hard to bend or compress in that direction. Or maybe the tension settles out very quickly because the glue they use has flow properties. (why do some speakers develop wrinkles, and why do panels slowly slide down?) Whatever is the reason, I only look at the final assembled product to gauge the tension and not the assembly procedure.
To answer some questions:
The panel doesn't need any horizontal tension to operate - vertical tension alone can provide a restoring force to drag the film back to the center.
Rich: I don't distinguish between "none" and "very very small".