The odd-order and generation conjecture for the differential-operator algebra of a matrix weight

Let $W$ be the matrix weight under consideration, and let $\mathcal D(W)$ denote its algebra of differential operators. The paper constructs four differential operators $D_1,D_2,D_3,D_4$ of order two and has verified computationally that there are no operators of order three or order five. Odd-order and generation conjecture. There are no operators of odd order in $\mathcal D(W)$, and the second-order differential operators $D_1,D_2,D_3,D_4$ generate the algebra $\mathcal D(W)$. These claims extrapolate the computations excluding orders three and five and the observed generation of the order-four quotient; their general validity is left open.