The stabilized inscribed-square conjecture for immersed plane curves

Let $T$ be an immersed curve in the plane, and let $St$, $J^+$, and $J^-$ denote its associated invariants. An inscribed square of a curve is a square whose vertices lie on the curve. Inscribed-square conjecture. There is a positive integer $m$ such that, for every $n\geq m$, there is an immersed curve $T_n$ with the same values of $St$, $J^+$, and $J^-$ as $T$, such that $T_n$ has exactly $n$ inscribed squares. Moreover, there is an integer $k$, independent of $n$, such that all but $k$ of the inscribed squares of $T_n$ have their vertices appearing in the same order as on $T_n$. The conjecture proposes that the number of inscribed squares can be made arbitrarily large while preserving these invariants, with all but uniformly boundedly many squares respecting the curve's cyclic order.