  Well-known theorems in classical analysis tell us that a pair $p_0$, $p_1$ of
real polynomials with positive coefficients have simple, interlacing roots if
and only if all nontrivial minors of a certain infinite matrix associated to
$p_0$ and $p_1$ -- called the Hurwitz matrix $H(p_0,p_1)$ -- are strictly
positive. What happens if we relax one side of this equivalence, allowing $p_0$
and $p_1$ to weakly interlace, i.e. have shared and/or repeated roots? It has
been shown in this case that $H(p_0,p_1)$ is still totally nonnegative. In the
present work, we prove the converse statement: if $H(p_0,p_1)$ is totally
nonnegative, then $p_0$ and $p_1$ weakly interlace. Thus our result completes
the analogy between the strict and weak versions of this equivalence. The proof
employs Temperley-Lieb immanants, first introduced by Rhoades and Skandera.
