The kernel decomposition conjecture for braided Wick operators

Let $\mathcal{H}$ be the Hilbert space underlying a Wick algebra, let $T$ be a braided operator on $\mathcal{H}^{\mathcal{\otimes}2}$, and let $R_m$ denote the corresponding Wick symmetrizer on $\mathcal{H}^{\mathcal{\otimes}m}$. Write $T_i$ for the copy of $T$ acting in tensor positions $i$ and $i+1$, and let $\mathcal{H}^{\mathcal{\otimes}0}$ be the scalar space. Kernel decomposition conjecture. If $T$ is braided, then

$$\ker R_{n+1}=(\mathbf{1}_{\mathcal{H}^{\otimes( n+1)}}-T_1T_2\cdots T_n)(\ker R_n\otimes\mathcal{H})+\ker R_{n-2}\otimes\ker R_2.$$

The conjecture would give a recursive description of the kernels of the Wick symmetrizers and hence of homogeneous Wick ideals. The source reports that direct Mathematica calculations support it for several examples, including Wick versions of the CCR, twisted CCR, twisted CAR, and quonic commutation relations; it is presented as unproved in the supplied text.