This is a lemma I wanted in order to solve Patrizio Neff's conjecture. It turned out to be the wrong way to think about it, but I am still curious if it is true.

Let $z^n+a_{n-1} z^{n-1} + \cdots + a_1 z + a_0$ and $z^n + b_{n-1} z^{n-1} + \cdots + b_1 z + b_0$ be two polynomials all of whose roots are real, satisfying $a_k \geq b_k > 0$ for $1 \leq k \leq n-1$ and $a_0=b_0 > 0$. Is there a family of polynomials $c(t)(z) = z^n + c_{n-1}(t) z^{n-1} + \cdots + c_1(t) z + c_0(t)$ such that $c(t)$ has real roots for all $t$, and the function $c_k$ decreases monotonically from $c_k(0)=a_k$ to $c_k(1)=b_k$?