Ghosh's regular-sequence conjecture for the Casas–Alvero problem

Let $K$ be a field, let $n$ be a positive integer, and set $R=K[x_1,\ldots,x_{n-1}]$. For $j\in\{1,\ldots,n-1\}$, let $\Phi_j:R\to R$ be given by $\Phi_j(x_i)=x_i-x_j$ for $i\neq j$ and $\Phi_j(x_j)=-x_j$, while $\Phi_n$ is the identity. Let $\sigma_i(x_1,\ldots,x_{n-1})$ be the $i$-th elementary symmetric function, set $\mathcal T=\{1,\ldots,n\}^{n-1}$, and for $T=(j_1,\ldots,j_{n-1})\in\mathcal T$ define $G_{T,i}=\Phi_{j_i}(\sigma_i(x_1,\ldots,x_{n-1}))$. Ghosh's regular-sequence conjecture. For every choice of $T\in\mathcal T$, the sequence of homogeneous polynomials

$$(G_{T,1},\ldots,G_{T,n-1})$$

forms a regular sequence in $R$. This is an equivalent algebraic formulation of the Casas–Alvero conjecture over arbitrary fields. The paper attributes this formulation to Soham Ghosh; its general validity remains open.