Balanced normal bundle conjecture for rational normal curves on Fano intersections of quadrics

Let $R_n\subset\mathbb{P}^n$ be the rational normal curve, and let $X\subset\mathbb{P}^n$ be a general Fano complete intersection of $k$ quadric hypersurfaces containing $R_n$. The normal bundle of $R_n$ on $X$ is denoted $N_{R_n/X}$; it is balanced when its summands on $\mathbb{P}^1$ have degrees differing by at most one. Balanced normal bundle conjecture. The normal bundle $N_{R_n/X}$ is balanced. The claim is presented as a natural conjecture following several proved cases for complete intersections of quadrics, but the supplied text gives no resolution beyond those cases.