Conjecture on diagonal-flip stabilization of surface triangulations

For a surface $S$, let $N(S)$ be the minimum value such that any two triangulations $T$ and $T'$ of $S$ with

$$|V(T)|=|V(T')|\geq N(S)$$

are equivalent under diagonal flips. Let $V_{\min}(S)$ be the minimum number of vertices in a triangulation of $S$.

Conjecture on diagonal-flip stabilization. The only surfaces $S$ for which

$$N(S)=V_{\min}(S)$$

are $S_0$, $S_1$, $S_2$, $N_1$, $N_2$, and $N_3$.

The equality means that flip-equivalence already holds at the minimum triangulation size. It is verified for the listed surfaces, while the paper gives counterexamples showing strict inequality for many higher-genus surfaces; the full exclusivity claim is left as a conjecture.