First-step cup-product dimension conjecture

Let $X$ be a $2$-dimensional simplicial complex, let ${\mathcal{F}}$ be a sheaf on $X$, and use the notation of the modification process. Set

$$M=\frac{|X(2)|-|X(1)|+|X(0)|-1}{2|X(2)|-|X(1)|+1}\dim{\mathcal{F}}-\bigl(h^1({\mathcal{F}})-h^0({\mathcal{F}})+1\bigr).$$

Assume $M\geq 0$. First-step cup-product dimension conjecture. With probability $1-o(1)$ as a function of $M$ (respectively, $|\mathbb{F}|$),

$$\dim E'_1=\dim\ker\left({\mathrm{H}}^1(X,\mathbb{F})\otimes_{\mathbb{F}}E'_0\longrightarrow {\mathrm{H}}^2(X,{\mathcal{F}}),\quad [\alpha]\otimes f\longmapsto[\alpha\cup f]\right).$$

The conjecture is presented as a consequence of the authors' simulations and is a special case of the broader dimension predictions for the modification process.