The generalized Jacobian conjecture on algebraic closedness

Let $k$ be a field of characteristic $0$, let $n>1$, and let $r\in\{2,\dots,n\}$. For polynomials $f_1,f_2,\dots,f_r\in k[x_1,\dots,x_n]$, write $jac^{f_1,\dots,f_r}_{x_{j_1},\dots,x_{j_r}}$ for the corresponding $r\times r$ Jacobian minors. The generalized Jacobian conjecture. If

$$\gcd\left(jac^{f_1,f_2,\dots,f_r}_{x_{j_1},x_{j_2},\dots,x_{j_r}}:1\leq j_1<\dots<j_r\leq n\right)\in k\setminus\{0\},$$

then $k[f_1,\dots,f_r]$ is algebraically closed in $k[x_1,\dots,x_n]$. This generalizes the Jacobian conjecture from full-dimensional polynomial maps to subalgebras generated by fewer polynomials; its general resolution is not known.