Consider Milnor's immersion of a circle in the plane which bounds two "incompatible" immersed disks [1]. I think I can glue these two disks together to define an immersion $f:S^2\to\mathbb{R}^3$ where Milnor's immersed circle sits in the $xy$-plane and contracts to a point (in the $\pm z$ direction) by running along either immersed disk.

Does $f(S^2)$ bound an immersed ball, i.e. does $f$ extend to an immersion of a 3-ball?

This question was sparked by glancing at Eliashberg-Mishachev’s paper “Topology of spaces of S-immersions” (and Gromov's book [1]), which to my understanding, argues that the projection $S^2\to\mathbb{R}^2$ (composing $f$ above with projection onto $xy$-plane) is not homotopic through folded maps to the "standard" folded map $(x,y,z)\mapsto (x,y)$ on the unit 2-sphere.

Milnor surprisingly found an immersion of a circle in the plane which bounds two "incompatible" immersed disks (see [1] for example, picture included). I think I can glue these two disks together to define an immersion of a sphere $f:S^2\to\mathbb{R}^3$ where Milnor's immersed circle (the equator) sits in the $xy$-plane and contracts to a point (in the $\pm z$ directions) by running along either immersed disk.

Does $f(S^2)$ bound an immersed ball, i.e. does $f$ extend to an immersion of a 3-ball?

This question was sparked by glancing at Gromov's book [1] and a related paper of Eliashberg-Mishachev which argues that the projection $S^2\to\mathbb{R}^2$ (composing $f$ above with projection onto $xy$-plane) is not homotopic through folded maps to the "standard" folded map $(x,y,z)\mapsto (x,y)$ on the unit 2-sphere (the fold being the equator).