Vector-valued Maclaurin inequality

Fix vectors $v_1,\dots,v_m\in\mathbb{R}^d$ with $1\le d\le m$. For integers $k$ satisfying $2\le k\le d$, let $|v_{i_1}\wedge\cdots\wedge v_{i_k}|$ denote the Euclidean norm of the exterior product. Vector-valued Maclaurin inequality. For every $p\in[0,\infty]$, one has

$$\left(\frac{\sum\limits_{1\le i_1<\cdots<i_k\le m}|v_{i_1}\wedge\cdots\wedge v_{i_k}|^p}{\binom{m}{k}}\right)^{\frac{1}{kp}} \le \left(\frac{\sum\limits_{1\le i_1<\cdots<i_{k-1}\le m}|v_{i_1}\wedge\cdots\wedge v_{i_{k-1}}|^p}{\binom{m}{k-1}}\right)^{\frac{1}{(k-1)p}},$$

with equality if and only if $m=d$ and the vectors $v_i$ form an orthonormal basis. Here the cases $p=0$ and $p=\infty$ are understood through the corresponding limiting interpretations of the displayed expression. Earlier results establish the inequality for $p=1$ in several cases, including $k=2,3,d$ in all dimensions when $m=d$, and $k=4$ in dimensions $5$ and $6$; the conjecture asserts the full range of exponents and parameters.