Hyper-Tauberian algebras defined by a Banach algebra homomorphism

Let $A$ and $B$ be Banach algebras and $T:B\longrightarrow A$ be a continuous homomorphism. We consider left multipliers from $A\times_T B$ into its the first dual i.e., $A^*\times B^*$ and we show that $A\times_T B$ is a hyper-Tauberian algebra if and only if $A$ and $B$ are hyper-Tauberian algebras.