Laguerre operators preserve the positive shifted Laguerre–Pólya class

Let $\phi(x)$ belong to the positive shifted Laguerre–Pólya class $\mathcal{SL\text{-}P}^{+}$, and let $L_j$ be the Laguerre operator defined by the coefficient expansion

$$|\phi(x+iy)|^2=\sum_{k=0}^{\infty}L_k(\phi(x))y^{2k}.$$

The Laguerre-operator preservation conjecture. For every $j\in\mathbb{N}$,

$$L_j(\phi(x))\in\mathcal{SL\text{-}P}^{+}.$$

This is the function-theoretic analogue of the preceding Turán-transform conjecture and is presented as an open conjecture.