Conjecture on hyperbolicity preservation by powers of the Legendre differential operator

Let $n$ be a positive integer and let $k$ be a positive integer with $k\leq n-1$. Define

$$\delta=(x^2-1)D^2+2xD.$$

Hyperbolicity-preservation conjecture. The operator

$$\delta^{n-k}(\delta^k-2^k)$$

is hyperbolicity preserving.

This conjecture proposes a family of hyperbolicity-preserving operators built from the Legendre differential operator. The source gives it as a further conjecture motivated by the symbol-curve analysis, with no proof or resolution stated.