Critical-threshold conjecture for the zeros of partial sums

Fix $N>0$. The polynomial $q_N(x;t)$ is the $N$th partial sum associated with the orthogonal-polynomial family in the paper.

Critical-threshold conjecture. There exists a threshold $t_{\textrm{crit}}=t_{\textrm{crit}}(N)>0$ such that $x\mapsto q_N(x;t)$ has $N$ real simple zeros for $t>t_{\textrm{crit}}$, while it has at least one pair of complex conjugate zeros for $0<t<t_{\textrm{crit}}$.

The preceding results establish the real simplicity of the zeros for sufficiently large $t$, while the discussion of the limit $t\to0$ shows that the zeros escape to infinity; the conjecture asserts a single positive threshold separating these two regimes. The supplied text gives no resolution of this threshold claim.