Wronskian positivity conjecture for totally nonnegative and totally positive spaces

Let $V\in\operatorname{Gr}_{k,n}(\mathbb{R})$. Regard the Wronskian $\operatorname{Wr}(V)$ as having a zero at $-\infty$ when its degree is less than $k(n-k)$. Wronskian positivity conjecture. If all complex zeros of $\operatorname{Wr}(V)$ lie in $[-\infty,0]$, then $V$ is totally nonnegative; if all lie in $(-\infty,0)$, then $V$ is totally positive. The nonnegative case was independently posed by Evgeny Mukhin and Vitaly Tarasov in 2017. The conjecture is a dual formulation of the osculating special case of the totally positive secant conjecture and remains open.