Completed projectWe are looking to build our own barn doors (2400H X 1200W) for 2 large openings in our living area. The design below is what we want to achieve.

We had originally planned on using individual pine planks attached to a thick piece of MDF, but to save time + money, we are now thinking of sandwiching together a piece of 9mm MDF Board with 2 pieces of 9mm 2 x 2400 x 1200 x 9mm V Grooved Project Panels (links to products at end of post). We would glue and then nail the V grooved panels to both sides of the MDF so we get the desired look on both sides of the door.
We would then screw in a horizontal 1200W X 235mm x 19mm Pine panel at the top and bottom of the door to add further stability and to attach the tracking/rail system to. This would bring the depth of the door to 46mm - which seems to be around the standard depth of a normal door.
My concern is whether the door will be too ‘flimsy’ as it will essentially be made out of 3 large 9mm boards. I will also be white washing and then sealing the door to help keep moisture out (even though it isn’t in a wet area).
What do you think? Do you think the door will hold up fine as described? Or will we need to attach another pine plank across the door to provide extra support? Something like this:

V grooved : https://www.bunnings.com.au/2400-x-1200-x-9mm-internal-v-grooved-project-panel_p0390169
MDF board : https://www.bunnings.com.au/9mm-standard-mdf-1200mm-x-2400mm_p0590056
cheers