Secant-variety extremality conjecture for blow ups along rational normal curves

Let $C\subset\mathbb{P}^n$ be a rational normal curve, and let $X_n=\operatorname{Bl}_C\mathbb{P}^n$. For an integer $d$, write $\operatorname{Sec}_d(C)$ for the $d$-secant variety of $C$, and let $\operatorname{Eff}_k(X_n)$ denote the cone of effective $k$-dimensional cycles on $X_n$. The proper transform of the cone over $\operatorname{Sec}_d(C)$ with vertex a point of $C$ is a subvariety of $X_n$. Secant-variety extremality conjecture. For every $2\leq d\leq n/2$, the proper transform of $\operatorname{Sec}_d(C)$ generates an extremal ray of $\operatorname{Eff}_{2d-1}(X_n)$. For every $1\leq d<n/2$, the proper transform of the cone over $\operatorname{Sec}_d(C)$ with vertex a point of $C$ generates an extremal ray of $\operatorname{Eff}_{2d}(X_n)$. This conjecture extends the established extremality results for the divisor, curve, surface, and relevant three-dimensional cones considered in the paper; the general cases remain open.