Standard-form classification conjecture for tau-symmetric RH perturbations

Let a $\tau$-symmetric Hamiltonian perturbation of the Riemann--Hopf hierarchy be written in generalized standard form, with first Hamiltonian

$$\widetilde{H}_1=\int\left(\frac{w^3}{6}-\epsilon^2\frac{a_0}{24}w_1^2+\sum_{k\geq 4}\epsilon^k\sum_{\substack{\mu\in\mathcal{P}_k,\,l(\mu)\geq 2\\ \mu_1=\mu_2,\,m_1(\mu)=0}}\alpha_\mu w_\mu\right)dx.$$

Here $\mathcal{P}_k$ denotes the set of partitions of $k$, $l(\mu)$ their length, $m_1(\mu)$ the multiplicity of $1$, and $w_\mu:=\prod_i w_{\mu_i}$. Standard-form classification conjecture. If $a_0=0$, then $\alpha_\mu=0$ for all $\mu$. If $a_0\neq0$, then $a_0,\alpha_{(2^2)},\alpha_{(2^3)},\ldots$ uniquely determine all other $\alpha_\mu$, all $\alpha_\mu$ vanish when $|\mu|$ is odd, and arbitrary values of $a_0\in\mathbb{C}^*$ and $\alpha_{(2^2)},\alpha_{(2^3)},\ldots\in\mathbb{C}$ are allowed. The supplied text gives this as a conjectural statement, with no resolution evidence.