Conjecture that every non-special plane system admits a reduction algorithm

Let $D\subset\mathbb{N}^{2}$ be finite, and consider a system $\mathcal{L}_{D}(m_{1},\dots,m_{r})$. The system admits a reduction algorithm if there is a sequence of sets

$$D=D_{r}\supset\dots\supset D_{0}$$

that satisfies the four reduction conditions in the source: the successive cardinality differences are bounded by $\binom{m_{j}+1}{2}$, the systems on the successive differences are non-special, the specified equal-sum subsets give special systems, and $\#D_{0}-1\leq\operatorname{edim}\mathcal{L}_{D}(m_{1},\dots,m_{r})$. Reduction-algorithm conjecture. Every non-special system admits a reduction algorithm. This would make the reduction method available for all non-special systems, complementing the theorem that any system admitting such an algorithm is non-special.