Generalized Cauchy–Schwarz inequality for real powers

Let $n$ be a positive integer, let $p\geq 2$ be a real number, and let $\mathbf{v},\mathbf{w}\in\mathbb{R}_{>0}^n$. For a vector, write $\mathbf{v}^p$ for the entrywise $p$-th power, and let $\langle\cdot,\cdot\rangle$ and $\|\cdot\|$ denote the Euclidean inner product and norm, respectively.

Generalized Cauchy–Schwarz conjecture. For every such $p$, $\mathbf{v}$, and $\mathbf{w}$,

$$\|\mathbf{v}^p\|\|\mathbf{w}^p\|-\langle\mathbf{v}^p,\mathbf{w}^p\rangle\leq \|\mathbf{v}\|^p\|\mathbf{w}\|^p-\langle\mathbf{v},\mathbf{w}\rangle^p.$$

The inequality is known for positive integer $p$ and fails for $p\in(0,1)$ and $p\in(1,2)$. Extensive numerical testing for non-integer $p\in(2,10)$ and dimensions $2\leq n\leq 10$ found no counterexamples, but the assertion for non-integer $p\geq2$ remains open; it is tight when $\mathbf{v}=\mathbf{w}=\mathbf{e}_1$.