The continuity conjecture for the optimal negative coefficient bound

Let $\mathcal{P}_n$ be the set of polynomials preserving nonnegative matrices of order $n$. For $m,s\in\mathbb{Z}^{\geq 0}$ with $s>m\geq n$, define

$$p_t(x)=\sum_{k=0}^{n-1}x^k-tx^m+\sum_{k=0}^{n-1}x^{s+k}.$$

Continuity conjecture. The largest $t\in\mathbb{R}^+$ such that $p_t\in\mathcal{P}_n$ for all such $m,s$ is a continuous function of $m$ and $s$, bounded between $1$ and $2$. The claim proposes a precise description of how the optimal coefficient bound varies with the exponents.