Boris Shapiro's total reality conjecture for convex curves

Let $\gamma:\mathbb{R}\to\mathbb{R}_{m-1}[u]$ be a convex curve, meaning that any $m$ distinct points along $\gamma$ are linearly independent. Let $z_1,\dots,z_{d(m-d)}$ be distinct real numbers, and for each $i$ let $W_i\in\mathrm{Gr}(m-d,d)$ be the osculating $(m-d)$-plane to $\gamma$ at $z_i$. Boris Shapiro's total reality conjecture for convex curves. There are exactly \mathsf{f}^{{\mathchoice% {\scalebox{1.35}{\raisebox{-0.02em}{\sqsubset!!\sqsupset}}} {\scalebox{1.35}{\raisebox{-0.02em}{\sqsubset!!\sqsupset}}} {\sqsubset\!\!\sqsupset} {\sqsubset\!\!\sqsupset} }} distinct solutions to the corresponding Schubert problem, with \nu={\mathchoice% {\scalebox{1.35}{\raisebox{-0.02em}{\sqsubset!!\sqsupset}}} {\scalebox{1.35}{\raisebox{-0.02em}{\sqsubset!!\sqsupset}}} {\sqsubset\!\!\sqsupset} {\sqsubset\!\!\sqsupset} }, and all solutions are real. This generalizes the Shapiro–Shapiro conjecture from the moment curve to arbitrary convex curves; the supplied text does not state its resolution.