The global extreme-zero product conjecture for transitional Laguerre polynomials

For $n=0,1,\dots$ and $\alpha>-1$, define the transitional Laguerre polynomials $\widehat{L}^{(\alpha)}_{n+1}(\cdot,t)$ by

$$\widehat{L}^{(\alpha)}_{n+1}(\cdot,t)=(x-(2n+\alpha+1))L^{(\alpha)}_n(x)-n(n+\alpha)t^2L^{(\alpha)}_{n-1}(x),$$

where

$$L^{(\alpha)}_{-1}=0,\qquad \widehat{L}^{(\alpha)}_n=(-1)^n n!L^{(\alpha)}_n.$$

Let the extreme zeros denote the smallest and largest zeros of a polynomial. Global extreme-zero product conjecture. For all $\alpha>-1$, $n=5,6,\dots$, and $t\in(0,1]$, the product of the extreme zeros of $\widehat{L}^{(\alpha)}_{n+1}(\cdot,t)$ is greater than the product of the extreme zeros of $\widehat{L}^{(\alpha)}_{n+1}(\cdot,0)$.

This extends the proposition's partial result, which covers only $-1<\alpha\leq47.9603$ and $t\in[0,0.568774]$. Numerical evidence includes the Hermite-related cases $\alpha=\pm1/2$ and $t=1$, but the assertion for all admissible $\alpha$, all $n\geq5$, and $t\in(0,1]$ remains open.