Convergence of Taylor series in Hardy norm

If $0< p<\infty$, let $H^p(D)$ denote the Hardy space on the unit disk. We know from Hardy spaces theory that, for every $f\in H^p(D)$, the family of functions $$f_r :D\rightarrow \mathbb{C}, z\mapsto f(rz)$$ converges in $H^p(D)$ to $f$ for $r\rightarrow 1^-$, basically stating that the the Taylor series of $f$ is Abel-summable in the $H^p(D)$ norm to $f$.

Moreover, as a corollary of Riesz theorem, for every $p\in(1,+\infty)$ and for every $f\in H^p(D),$ we can deduce that the Taylor series of $f$ converges in the $H^p(D)$ norm to $f$.

So the question: for which $0<p\le1$ is it true that, for all $f\in H^p(D)$, the Taylor series of $f$ converges in the $H^p(D)$ norm to $f$? Has this question a known answer? Can someone give me any literature reference?

• Ever thought of looking at something simple like $\frac 1{1-z}$ for $p<1$? – fedja Jun 21 '18 at 7:13
• Thanks a lot for the suggestion: I've just done the calculations and the counterexample works. If you want to write it as an answer, I'll be happy to give you the bounty :) Do you also know anything about $p=1$? – Bob Jun 21 '18 at 7:52

As pointed out by fejia in the comments, the function $$D\rightarrow\mathbb{C},z\mapsto\frac{1}{1-z}$$ provides a counter-example for the cases $0<p<1$.
For the case $p=1$, the answer is negative too: a detailed exposure can be found in the article "Kehe Zhu - Duality of Bloch spaces and norm convergence of Taylor series".