6. Orthogonal polynomials

Let $(L_n^\alpha)$ denote the generalized Laguerre polynomials. It is known that $L_m^\alpha$ and $L_n^\alpha$ can have common zeros if $mn>1$. Is the number of common zeros linked to arithmetic properties of $\alpha$?
Conjecture 6.1.
[K. Driver] There are no common zeros between $L_m^\alpha$ and $L_n^\alpha$, $mn>1$, for $\alpha$ rational. 
Problem 6.2.
[K. Driver] Let $P_n^{\;\alpha, \,\beta}$ denote the Jacobi polynomial of $\deg n$ with parameters $\alpha, \beta> 1$. Then $P_n^{\;\alpha', \,\beta'}$ and $P_n^{\;\alpha, \,\beta}$ interlace for $\alpha\approx\alpha'$, and $\beta\approx\beta'$. Does the failure of interlacing correspond to some physical law being “violated”? 
Problem 6.3.
[P. Moussa] Consider a subspace $S$ of $\mathbb{C}[x]$ of codimension $k$ (defined by $k$ linear constraints, explicitly given). Given a scalar product on $\mathbb{C}[x]$, find an orthogonal basis of $S$ (ordered by degree); some degrees will be missing. Which degrees get skipped?
 Characterize these orthogonal polynomials.
 Does this have anything to do with lacunary Padé approximants?
Cite this as: AimPL: Stability and hyperbolicity, available at http://aimpl.org/hyperbolicpoly.