The bilinear-equation conjecture for the Boussinesq polynomials

Let $\Theta_n^{\pm}(x,t)$ be the polynomials defined by

$$\Theta_n^{\pm}(x,t)=xP_n(x,t)\pm \operatorname{i}tQ_n(x,t),$$

where $P_n(x,t)$ and $Q_n(x,t)$ are the polynomials in the generalised rational solution of the Boussinesq equation. Bilinear-equation conjecture. The polynomials $\Theta_n^{\pm}(x,t)$ satisfy the bilinear equation

$$\text{the bilinear equation defining the Boussinesq rational solutions}.$$

This has been verified for $n=1,2,3,4$ in the paper; the general case is conjectural.