Common-orbit conjecture for real polynomials under differential operators

Let $\mathcal{R}_{n}$ be the set of all monic real polynomials of degree $n$, and let $\mathcal{S}$ be the semigroup of differential operators defined in the paper. For $P_{1},P_{2}\in\mathcal{R}_{n}$, the conjecture concerns whether their $\mathcal{S}$-orbits intersect.

Common-orbit conjecture. If $P_{1},P_{2}\in\mathcal{R}_{n}$, then there exist differential operators $\Lambda_{1},\Lambda_{2}\in\mathcal{S}$ such that

$$\Lambda_{1}P_{1}=\Lambda_{2}P_{2}.$$

This asserts that any two $\mathcal{S}$-orbits in $\mathcal{R}_{n}$ have a non-empty intersection. The source presents it as a belief arising in its discussion of the semigroups $\mathcal{S}$ and $\widetilde{\mathcal{S}}$, with no resolution supplied.