There is a theorem by Mooney http://msp.org/pjm/1972/43-2/pjm-v43-n2-p.pdf#page=185 and independently proved by Havin which says that the predual of $H^{\infty}(\mathbb{D}),$ $L^{1}/H^{1}_{0}$ is weakly sequentially complete, i.e., to say if we have a sequence of weak* continuous linear functionals $\lbrace\Lambda_n;n\in\mathbb{Z}_{+}\rbrace$ such that $\Lambda\varphi:=\lim\Lambda_n\varphi$ exists for each $\varphi\in H^{\infty}(\mathbb{D}).$ Then $\Lambda$ is weak* continuous. I have following questions:

is the conclusion still holds true if the sequential limit exists apriori only for $\varphi\in A(\mathbb{D})?$

or if there are some results about the maximal set where if the sequential limit exits the conclusion of the above theorem still holds?

Remark: $H^{\infty}(\mathbb{D})$ and $A(\mathbb{D})$ respectively represents the Banach spaces of bounded analytic functions and the holomorphic functions which are continuous upto the boundary of $\mathbb{D}.$