Monotonicity conjecture for infinite-order hyperbolicity-preserving operators

Let $T=\sum_{k=0}^{\infty}T_k(x)D^k$ be an infinite-order differential operator on $\mathbb{R}[x]$, where $D$ denotes differentiation, and call $T$ monotone if $\deg T_k\leq\deg T_{k+1}$ whenever $\deg T_k\geq 0$. The operator $T$ is hyperbolicity preserving if it maps every real-rooted polynomial to a real-rooted polynomial or zero.

Monotonicity conjecture. If $T$ is not monotone, then $T$ is not hyperbolicity preserving.

This conjecture relates the degrees of the coefficient polynomials in an infinite-order differential operator to preservation of hyperbolicity. It was posed in the study of Legendre multiplier sequences; the supplied source does not establish whether it has been resolved.