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There is no such sequence.

For an immersion $f_k\colon \mathbb S^2 \rightarrow \mathbb{R}^3$ (after a small perturbation) the set of self-intersections is formed by some number of closed curves $\gamma_1,,\gamma_2,\dots \gamma_n,$ in $\mathbb R^3$. So any plane which intercets all $\gamma_i$ transversally, has to intersect them at even number of points.

On the other hand the the equator plane say $\Pi$ (or its small perturbation) has to itersect it odd number of times. Indeed, a contradictionthe curves in $f_k^{-1}(\Pi)$ is close to equator $\mathbb S^2$; the turning number of its image in $\Pi$ is $2$; so it has odd number of self-intersections. (This works for if $f_k$ is $C^1$-close to $z^2$ near $\Pi$, which is easy to arrange.)

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There is no such sequence.

For an immersion $f_k\colon \mathbb S^2 \rightarrow \mathbb{R}^3$ the set of self-intersections is formed by some number of closed curves $\gamma_1,,\gamma_2,\dots \gamma_n,$ in $\mathbb R^3$. So any plane which intercets all $\gamma_i$ transversally, has to intersect them at even number of points.

On the other hand the the equator plane (or its small perturbation) has to itersect it odd number of times, a contradiction.