Convolution operator on $L_\infty(\ell_q)$

Question

Let $\mathcal{S}(\mathbb{R}^{n})$ be the Schwartz space of rapidly decreasing functions on $\mathbb{R}^{n}$ and let $\varphi \in \mathcal{S}(\mathbb{R}^{n})$. I would like to know whether the convolution operator \begin{align*} T_{\varphi}((f_{k})_{k=0}^{\infty}) = (\varphi * f_{k})_{k=0}^{\infty} \end{align*} is bounded from $L_{\infty}(\ell_{q})$ into $L_{\infty}(\ell_{q})$ for $ 0 < q < 1$. Here \begin{align*} L_{\infty}(\ell_{q}) := \{(f_{k})_{k=0}^{\infty}: \mathbb{R}^{n} \rightarrow \mathbb{C} \text{ Lebesgue measurable }: \|(f_{k})_{k=0}^{\infty}|L_{\infty}(\ell_{q})\|=\|(\sum_{k=0}^{\infty} |f_{k}(x)|^{q})^{1/q}|L_{\infty}(\mathbb{R}^{n})\| < \infty\}. \end{align*} I know this is true for $ 1 \leq q < \infty$ even if we just assume $\varphi \in L_{1}(\mathbb{R}^{n})$ instead of $\varphi \in \mathcal{S}(\mathbb{R}^{n})$. The proof is just based on Hölder inequality and the monotone convergence theorem: \begin{align*} \|T_{\varphi}(f_{k})|L_{\infty}(\ell_{q})\| &= \text{ess }\sup_{x \in \mathbb{R}^{n}} (\sum_{k=0}^{\infty} |\varphi * f_{k}(x)|^{q})^{1/q} = \text{ess }\sup_{x \in \mathbb{R}^{n}} \Bigg(\sum_{k=0}^{\infty} \Bigg| \int_{\mathbb{R}^{n}}\varphi(x-y)f_{k}(y) dy\Bigg|^{q}\Bigg)^{1/q} \\ &\leq \text{ess }\sup_{x \in \mathbb{R}^{n}}\Bigg(\sum_{k=0}^{\infty} \Bigg(\int_{\mathbb{R}^{n}}|\varphi(x-y)|^{1-1/q}|\varphi(x-y)|^{1/q}|f_{k}(y)| dy\Bigg)^{q}\Bigg)^{1/q}\\ &\leq \text{ess }\sup_{x \in \mathbb{R}^{n}} \Bigg(\sum_{k=0}^{\infty}\Bigg(\int_{\mathbb{R}^{n}} |\varphi(x-y)|dy\Bigg)^{q/q'}\Bigg(\int_{\mathbb{R}^{n}}|\varphi(x-y)||f_{k}(y)|^{q} dy\Bigg)\Bigg)^{1/q}\\ &=\|\varphi|L_{1}(\mathbb{R}^{n})\|^{1/q'}\cdot \text{ess }\sup_{x \in \mathbb{R}^{n}} \Bigg(\sum_{k=0}^{\infty}\int_{\mathbb{R}^{n}}|\varphi(x-y)||f_{k}(y)|^{q} dy\Bigg)^{1/q}\\ &=|\varphi|L_{1}(\mathbb{R}^{n})\|^{1/q'}\cdot \text{ess }\sup_{x \in \mathbb{R}^{n}} \Bigg(\int_{\mathbb{R}^{n}}|\varphi(x-y)|\sum_{k=0}^{\infty}|f_{k}(y)|^{q} dy\Bigg)^{1/q}\\ &\leq \|\varphi|L_{1}(\mathbb{R}^{n})\|^{1/q'}\|\varphi|L_{1}(\mathbb{R}^{n})\|^{1/q} \text{ess }\sup_{y \in \mathbb{R}^{n}} (\sum_{k=0}^{\infty} |f_{k}(y)|^{q})^{1/q} \\ &= \|\varphi|L_{1}(\mathbb{R}^{n})\| \|(f_{k})_{k=0}^{\infty}|L_{\infty}(\ell_{q})\| \end{align*} This proof does not work when $0 < q < 1$. Does anyone have any idea how to prove or disprove the statement for $0 < q < 1$?