Conjecture on the bispectral algebra for a three-by-three step-up example

Let

$$\Psi(x,z)=e^{xz}\begin{pmatrix}z-x^{-1}&x^{-2}&-x^{-3}\\0&z-x^{-1}&x^{-2}\\0&0&z-x^{-1}\end{pmatrix}.$$

Consider matrix-valued polynomials $\Theta(x)$ for which there exists a differential operator $\mathcal B$ satisfying

$$\Psi\mathcal B=\Theta(x)\Psi.$$

Bispectral algebra conjecture. The algebra of all such $\Theta(x)$ is the algebra of polynomials

\begin{aligned} &\begin{pmatrix}r^{11}_0&r^{12}_0&r^{13}_0\\0&r^{22}_0&r^{23}_0\\0&0&r^{11}_0\end{pmatrix}+\begin{pmatrix}r^{11}_1&r^{12}_1&r^{13}_1\r^{22}_0-r^{11}_0&r^{22}_1&r^{23}_1\\0&r^{22}_0-r^{11}_0&r^{11}_1+r^{23}_0-r^{12}_0\end{pmatrix}x\\ &+\begin{pmatrix}r^{11}_2&r^{12}_2&r^{13}_2\r^{22}_1-r^{11}_1-r^{23}_0+r^{12}_0&r^{22}_2&r^{23}_2\r^{22}_0-r^{11}_0&r^{22}_1-r^{11}_1&r^{11}_2+r^{23}_1-r^{12}_1\end{pmatrix}x^2\\ &+\begin{pmatrix}r^{11}_3&r^{12}_3&r^{13}_3\r^{21}_3&r^{22}_3&r^{23}_3\r^{22}_1-2r^{11}_1-r^{23}_0+r^{12}_0&r^{32}_3&r^{33}_3\end{pmatrix}x^3\\ &+\begin{pmatrix}r^{11}_4&r^{12}_4&r^{13}_4\r^{21}_4&r^{22}_4&r^{23}_4\r^{32}_3+r^{21}_3-r^{22}_2-r^{11}_2+r^{12}_1&r^{32}_4&r^{33}_4\end{pmatrix}x^4\\ &+\begin{pmatrix}r^{11}_5&r^{12}_5&r^{13}_5\r^{21}_5&r^{22}_5&r^{23}_5\r^{32}_4+r^{21}_4-r^{33}_3-r^{22}_3-r^{11}_3+r^{23}_2+r^{12}_2-r^{13}_1&r^{32}_5&r^{33}_5\end{pmatrix}x^5+x^6P(x), \end{aligned}

where $P(x)$ is an arbitrary $3\times3$ matrix-valued polynomial and all variables $r^{ij}_k$ displayed in the source are arbitrary.