Nonvanishing conjecture for extremals of $C(k,p)$

Let $0<p<1$, and let $f$ be an extremal for $C(k,p)$, where $C(k,p)$ denotes the coefficient-estimate constant studied in the paper. Nonvanishing conjecture. Any such extremal $f$ does not vanish in the unit disk $4\mathbb{D}4$. This conjecture is motivated by the known extremals for $C(1,p)$, $C(2,p)$, and $C(3,2/3)$, all of which do not vanish in $4\mathbb{D}4$. The paper states that it has little concrete evidence beyond these examples; if true, the result would reduce the proofs to the case $l=0$.