Homology bound for distinct split regions

For every field $K$, every finite-dimensional $K$-vector space $V$, every finite index type $\iota$ with a finite enumeration, and every family of finite-dimensional $K$-vector spaces $(W_i)_{i\in\iota}$, with each vector space given by an additive commutative group and a $K$-module structure, let $d:V\to V$ be $K$-linear with $d\circ d=0$, and let $u_i:W_i\to V$ and $p_i:V\to W_i$ be $K$-linear maps such that $p_i\circ u_i=\operatorname{id}_{W_i}$ for every $i$ and $p_i\circ u_j=0$ whenever $i\ne j$. For any functions $o,t:\iota\to\mathbb N$ such that $\dim_K\operatorname{im}(d\circ u_i)\leq o_i$ and $\dim_K\operatorname{im}(p_i\circ d)\leq t_i$ for every $i$, the inequality $\sum_{i\in\iota}\bigl((\dim_K W_i-o_i)-t_i\bigr)\leq\dim_K\bigl(\ker d/(\ker d\cap\operatorname{im}d)\bigr)$ holds. The intersection in the quotient is viewed inside $\ker d$ and equals $\operatorname{im}d$ because $d^2=0$. Both subtractions in each summand are successive natural-number subtractions, each truncated at zero, so that summand is $\max\{\dim_K W_i-o_i-t_i,0\}$ when the expression inside the maximum is read in the integers. The spaces and the bounds may be zero, and $\iota$ may be empty; in the empty case all indexed hypotheses are vacuous and the sum is zero.