Exponential concentration implies a finite Poincaré constant

Let $g(t)=\exp(t-e^t)$, and let $\gamma(x,\xi)=\exp(-a|x|-b|\xi|)$ with $a,b>0$. Suppose that $f\in L^2(\mathbb{R})$ has exponential time-frequency concentration, meaning

$$\exists\,\delta>0:\quad |\mathcal{V}_g f(z)|\cdot e^{\delta|z|}\in L^\infty(\mathbb{R}^2).$$

Let $w=(|\mathcal{V}_g f|^2\ast\gamma)^2$. Exponential-concentration conjecture. The weighted measure defined by $w$ has finite Poincaré constant:

$$C_P(w)<\infty.$$

Exponential concentration is known to follow from a Poincaré inequality, but the converse is not generally true for weights with disconnected support. In this setting, however, the weights are strictly positive because they arise from convolution with $\gamma$, so the question is whether exponential time-frequency concentration is sufficient for a finite Poincaré constant.