Regular-sequence conjecture for nearly commuting matrices

Let $X=(x_{st})$ and $Y=(y_{st})$ be $n\times n$ matrices of indeterminates, let $R=K[X,Y]$, and let $I$ be generated by the off-diagonal entries of $XY-YX$. Define

$$\theta(s,t)=\begin{cases}(s+t)\bmod n,&s+t\neq n,\\ n,&s+t=n.\end{cases}$$

Consider the elements

$$x_{st}-y_{t,\theta(s,t)},\quad x_{1n},\quad x_{nn},\quad x_{11}-y_{2n}.$$

Regular-sequence conjecture. The displayed collection is a regular sequence on $R/I$ and hence is part of a system of parameters on $R/J$ and $R/Q$.

The source reports verification by Macaulay2 for $n=3,4$ over $K=\mathbb{Q}$ and in some small prime characteristics. The general assertion remains open in the supplied account.