A counterexample concerning the coefficient formula for de Branges-Rovnyak norms

Let $\mathcal{H}(b)$ be the de Branges-Rovnyak space associated to a non-extreme point $b$ of the unit ball of $H^{\mathrm{\infty }}$, and let $\phi =b\mathrm{/}a$, where $a$ is the Pythagorean mate of $b$. In an earlier article, the author showed that, if $\phi \in H^2$, then $$f\in \mathcal{H}(b)⟺(f\in H^2\text{ and }\sum _{m\ge 0}\mid \sum _{n\ge 0}\overline{\widehat{\phi }(n)}\widehat{f}(m+n){\mid }^2<\mathrm{\infty }),$$ in which case we have $$\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathcal{H}(b)}^2=\sum _{m\ge 0}\mathrm{\mid }\widehat{f}(m){\mathrm{\mid }}^2+\sum _{m\ge 0}\mid \sum _{n\ge 0}\overline{\widehat{\phi }(n)}\widehat{f}(m+n){\mid }^2\mathrm{.}$$ It was also shown that there exists a space $\mathcal{H}(b)$ for which $\phi \in {\cap }_{p<2}{H}^p$, yet $$f\in \mathcal{H}(b)\Rightarrow ̸\sum _{m\ge 0}\mid \sum _{n\ge 0}\overline{\widehat{\phi }(n)}\widehat{f}(m+n){\mid }^2<\mathrm{\infty }\mathrm{.}$$ Here we complete this circle of ideas by showing that there exists a space $\mathcal{H}(b)$ for which $\phi \in {\cap }_{p<1}{H}^p$, yet $$(f\in H^2\text{ and }\sum _{m\ge 0}\mid \sum _{n\ge 0}\overline{\widehat{\phi }(n)}\widehat{f}(m+n){\mid }^2<\mathrm{\infty })\Rightarrow ̸f\in \mathcal{H}(b)\mathrm{.}$$