Lau–Ülger–Pym conjecture on factorization in Banach algebras

Let $A$ be a Banach algebra. It is faithful if its natural action on its dual is faithful, and it is weakly sequentially complete if every weakly Cauchy sequence in $A$ converges weakly in $A$. A bounded approximate identity (BAI) is a bounded net $(e_i)$ in $A$ such that $e_i a\to a$ and $a e_i\to a$ for every $a\in A$.

Lau–Ülger–Pym conjecture. There is no faithful infinite-dimensional non-unital weakly sequentially complete Banach algebra $A$ with a BAI such that

$$A^* = A^*A.$$

This concerns the relationship between factorization of the dual Banach algebra and structural properties such as unitality and weak sequential completeness. The source presents it as a conjecture attributed to Lau and Ülger and formulated by Pym; no resolution is given here.