Finite-dimensional characterization conjecture for spaces with complemented hyperplanes

Let $X$ be a finite-dimensional Banach space, with $n=\dim X$. Finite-dimensional characterization conjecture. (a) If every one-dimensional subspace of $X$ can be represented as an intersection of $1$-complemented hyperplanes, then $X$ is isometric to $\ell_p^n$ for some $1\leq p\leq\infty$. (b) There exists a finite-dimensional Banach space $X$ that is not isometric to $\ell_p^n$ for any $1\leq p\leq\infty$, but such that every $1$-complemented subspace $Y$ of $X$ with $\dim Y\geq 2$ can be represented as an intersection of $1$-complemented hyperplanes. The conjecture concerns whether the intersection property for complemented hyperplanes characterizes finite-dimensional $\ell_p^n$ spaces, while allowing the stated contrasting example in part (b). The source gives no resolution of either part.