Persistence-rank conjecture for symplectic rational surfaces

Let $X=\mathbb{C}P^2\# n\overline{\mathbb{C}P^2}$ carry a symplectic form $\omega$, and let $\Gamma(X,\omega)$ be its Lagrangian root system. Write $PR[\Gamma(X,\omega)]$ for the persistence rank, and let $\operatorname{Rank}[\pi_1(\operatorname{Symp}(X,\omega))]$ and $\operatorname{Rank}[\pi_0(\operatorname{Symp}_h(X,\omega))]$ denote the ranks appearing in the source. Persistence-rank conjecture. The quantity

$$PR[\Gamma(X,\omega)]+\operatorname{Rank}[\pi_1(\operatorname{Symp}(X,\omega))]-\operatorname{Rank}[\pi_0(\operatorname{Symp}_h(X,\omega))]$$

is constant and equals

$$1+2+\cdots+n=\frac{n(n+1)}{2}$$

for any symplectic form on $\mathbb{C}P^2\# n\overline{\mathbb{C}P^2}$. This conjecture relates the persistence type of the Lagrangian root system to homotopy and symplectic mapping-class-group data. The source gives the $n=6$ calculation as supporting evidence, with value $21$, but does not state a general proof or resolution.