Aleksandrov–Buryak–Tessler conjecture for the extended refined partition function

Let $N$ be any value for which the formal series are defined, let $\tau^{o,\mathrm{ext}}_N$ be the extended refined open partition function, and let $\tau_N$ be the Kontsevich–Penner tau-function. Use the formal variables $t_i$ and $s_i$ and make the substitutions

$$T_{2i+1}=\frac{t_i}{(2i+1)!!},\qquad T_{2i+2}=\frac{s_i}{2^{i+1}(i+1)!}.$$

Extended refined Kontsevich–Penner conjecture. For any $N$ we have

$$\tau^{o,\mathrm{ext}}_N=\left.\tau_N\right|_{\substack{T_{2i+1}=\frac{t_i}{(2i+1)!!},\T_{2i+2}=\frac{s_i}{2^{i+1}(i+1)!}.}}$$

The equality is known for $N=1$ and the source states that the conjecture is proved in genus $0$ and $1$; it remains open in general.