Family-size stabilization conjecture for complete multipartite graphs

Let $n \geq 3$, let $1 \leq a_1 \leq \cdots \leq a_n$, and define

$$e = \#E(K_{a_1, \ldots, a_{n-1}}).$$

If $a_1 + \cdots + a_{n-1} > 6$ and $a_n \geq e$, then the family-size stabilization conjecture asserts that

$$\left|\mathcal{F}(K_{a_1, \ldots, a_{n-1}, e-1})\right| = \left|\mathcal{F}(K_{a_1, \ldots, a_{n-1}, a_n})\right| - 1.$$

Here $K_{a_1,\ldots,a_n}$ is the complete multipartite graph with part sizes $a_1,\ldots,a_n$, $\#E(G)$ denotes the number of edges of $G$, and $\mathcal{F}(G)$ denotes the family of graphs obtainable from $G$ by sequences of Triangle-Y and Y-Triangle moves. The conjecture predicts a precise one-unit difference between the family sizes at the indicated parameters, extending the pattern observed in the tables and the proved stabilization result for the related family-size function. The general assertion remains open in the source.