The mu-class closure conjecture for rational parametrizations

Let ${\mathbb K}$ be an algebraically closed field. For a positive integer $n$, let ${\mathcal P}_n$ be the set of triples $(a,b,c)\in {\mathbb K}[t]_n^3$ with $\gcd(a,b,c)=1$ and $\deg(a,b,c)=n$, and let $\mu(a,b,c)$ be the least degree of a nonzero syzygy of $(a,b,c)$. Define

$${\mathcal P}^\mu_n:=\{(a,b,c)\in{\mathcal P}_n:\mu(a,b,c)=\mu\}.$$

Mu-class closure conjecture. For every $\mu\leq\lfloor n/2\rfloor$,

$$\overline{{\mathcal P}^\mu_n}={\mathcal P}^0_n\cup\dots\cup{\mathcal P}^\mu_n.$$

Equivalently, if $\mu<\lfloor n/2\rfloor$, every parametrization of class $\mu$ is a limit of parametrizations of class $\mu+1$. The result describes the closure relations among strata of rational parametrizations by syzygy class; the maximal-class closure was already known, while the asserted relations for lower classes were conjectural in the cited work.