But... I found a solution: make a double air exit, in order to flatten its two openings, and provide the required very thin air flow.
For that, I added a little flap on the small piece, folded it (the plastic quality has to stand with repeated foldings). This little flap is glued above the mouth hole: it keeps the airway cover at the right place and also saves the "flatability" of the nose flute.
For the rest, nothing hs changed, but it is easier now to show how this pop-up nose flute (2.5D!) works:
Here is the new template:
And the video!: