Existence of hamiltonian 2-form metrics on direct sum bundles

Let $B$ be a compact Kähler manifold and let $L\to B$ be a line bundle as above. Let $m_1,\dots,m_\ell$ be positive integers, $d_1,\dots,d_\ell$ nonnegative integers, and let $M$ be the total space of

$$E:=\bigoplus_{j=1}^\ell\left(\bigoplus_{k=0}^{d_j}L^{m_j}\right)\to B.$$

Existence conjecture. The conclusions of Theorem 1 hold for $M$, depending on the sign of $c_1(\det E\otimes K_B^{\vee})$. If $d_\ell=0$ and $\det E\otimes K_B^{\vee}\to B$ is trivial, then the conclusions of Theorem 2 hold on $M$. In both cases, the resulting metrics admit a hamiltonian 2-form of order $\ell$. The statement concerns extending the paper's explicit complete Ricci-flat metrics and Kähler-Ricci solitons from the previously treated bundles to these direct sum bundles; the surrounding text indicates that the remaining difficulties are essentially computational.