Brazitikos–McIntyre conjecture on vector Maclaurin inequalities

Let $x_1,x_2,\ldots,x_n\in\mathbb{R}^d$ be given with $1\leq d\leq n$. For $p\in[0,\infty]$ and $2\leq k\leq d$, consider the $k$-fold exterior products $x_{i_1}\wedge\cdots\wedge x_{i_k}$ and their Euclidean norms. Brazitikos–McIntyre conjecture. For every such $p$ and $k$, one has

$$\left(\frac{\sum_{1\leq i_1<\cdots<i_k\leq n}|x_{i_1}\wedge\cdots\wedge x_{i_k}|^p}{\binom{n}{k}}\right)^{\frac{1}{pk}} \leq \left(\frac{\sum_{1\leq i_1<\cdots<i_{k-1}\leq n}|x_{i_1}\wedge\cdots\wedge x_{i_{k-1}}|^p}{\binom{n}{k-1}}\right)^{\frac{1}{p(k-1)}},$$

with equality if and only if $n=d$ and the vectors form an orthonormal basis. This extends Maclaurin's inequality from positive scalars to norms of exterior products of vectors; its status is not resolved by the supplied source information.