An assembly (1) comprises: a first structure (10) formed by first filamentary elements (15), a second structure (12) formed by second filamentary elements (16), a supporting structure (14) comprising supporting filamentary elements linking the first filamentary elements (15) and the second filamentary elements (16), and at least one filamentary securing element (18) interlaced with the first filamentary elements (15) and the second filamentary elements (16), one of the ends (18a, 18b) of the securing element (18) being free to slip relative to the first and second filamentary elements (15, 16).

The tire (10) has a stiffening structure (50) comprising a stiffening element (52, 53) anchored in or around radially inner reinforcing structures (60A) and a radially outer reinforcing structure (70A, 70B). Each stiffening element (52, 53) extending from one of the radially inner reinforcing structures (60A) comprises a portion (523) extending in the toroidal cavity (35) between first anchor points. Each stiffening element (52, 53) extending from the other of the radially inner reinforcing structures (60B) comprises a portion (524) extending in the toroidal cavity (35) between second anchor points. The portions (523) do not cross over in the toroidal cavity (35). The first anchor points are axially on the other side of the same side of the median plane (M) of the tire (10) as the second anchor points.