The polynomial cluster value problem for Banach spaces

We reduce the polynomial cluster value problem for the algebra of bounded analytic functions, $H^{\mathrm{\infty }}$, on the ball of Banach spaces $X$ to the same polynomial cluster value problem for $H^{\mathrm{\infty }}$ but on the ball of those spaces which are ${\mathrm{\ell }}_1$-sums of finite dimensional spaces.