The positivity-induced triangle inequality for even-power forms

Let $q\geq 1$, and let $s_t en\mathbb{R}$ and $v_t\in\mathbb{R}^m$ for $t\in [w]$ satisfy

$$\sum_{t=1}^w s_t\langle v_t,x\rangle^{2q}\geq 0$$

for every $x\in\mathbb{R}^m$. Positivity-induced triangle inequality. For all $x,y\in\mathbb{R}^m$,

$$\left(\sum_{t=1}^w s_t\left(\langle v_t,x\rangle^2+\langle v_t,y\rangle^2\right)^q\right)^{1/q}\leq \left(\sum_{t=1}^w s_t\langle v_t,x\rangle^{2q}\right)^{1/q}+\left(\sum_{t=1}^w s_t\langle v_t,y\rangle^{2q}\right)^{1/q}.$$

This inequality would extend the preceding argument for the matrix Khintchine-type upper bound to decompositions in which the coefficients $s_t$ need not all be nonnegative, and would yield the cited upper-bound lemma.